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J. Reine Angew. Math. 169(158–176), 17–32 (1933)\nDeligne, P.: La conjecture de Weil pour les surfaces \\(K_3\\). Invent. Math. 15, 206–226 (1972)\nElliott, P.D.T.A.: Duality in Analytic Number Theory, vol. 122. Cambridge University Press, Cambridge (1997)\nElliott, P.D.T.A., Moreno, C.J., Shahidi, F.: On the absolute value of Ramanujan’s \\(\\tau \\)-function. Math. Ann. 266(4), 507–511 (1984)\nElliott, P.D.T.A., Kish, J.: Harmonic analysis on the positive rationals ii: Multiplicative functions and maass forms. preprint arXiv:1405.7132, (2014)\nGodement, R., Jacquet, H.: Zeta Functions of Simple Algebras. Lecture Notes in Mathematics, vol. 260. Springer, Berlin (1972)\nGelbart, S., Jacquet, H.: A relation between automorphic forms on \\({\\rm GL}(2)\\) and \\({\\rm GL}(3)\\). Proc. Natl. Acad. Sci. USA 73(10), 3348–3350 (1976)\nHafner, J.L., Ivic, A.: On sums of fourier coefficients of cusp forms. Enseign. Math. 35(2), 375–382 (1989)\nHecke, E.: Theorie der Eisensteinschen Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik. Abh. Math. Sem. Univ. Hamburg 5(1), 199–224 (1927)\nHenniart, G.: Sur la fonctorialité, pour \\(\\rm GL(4)\\), donnée par le carré extérieur. Mosc. Math. J. 9(1), 33–45 (2009)\nHolowinsky, R.: A sieve method for shifted convolution sums. Duke Math. J. 146(3), 401–448 (2009)\nJacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I. Am. J. Math. 103(3), 499–558 (1981)\nKim, H.H.: Functoriality for the exterior square of \\({\\rm GL}_4\\) and the symmetric fourth of \\({\\rm GL}_2\\). J. Am. Math. Soc. 16(1), 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak (2003)\nKim, H.H.: A note on Fourier coefficients of cusp forms on \\({\\rm GL}_n\\). Forum Math. 18(1), 115–119 (2006)\nKim, H.H., Shahidi, F.: Functorial products for \\(\\rm GL_2\\times GL_3\\) and functorial symmetric cube for \\(\\rm GL_2\\). C. R. Acad. Sci. Paris Sér. I Math. 331(8), 599–604 (2000)\nKim, H.H., Shahidi, F.: Cuspidality of symmetric powers with applications. Duke Math. J. 112(1), 177–197 (2002)\nKim, H.H., Shahidi, F.: Functorial products for \\({\\rm GL}_2\\times {\\rm GL}_3\\) and the symmetric cube for \\({\\rm GL}_2\\). Ann. Math. 155(3), 837–893. With an appendix by Colin J. Bushnell and Guy Henniart (2002)\nKloosterman, H.D.: Asymptotische formeln für die fourierkoeffizienten ganzer modulformen. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg vol. 5, pp. 337–352. Springer (1927)\nLiu, J., Wang, Y., Ye, Y.: A proof of Selberg’s orthogonality for automorphic \\(L\\)-functions. Manuscripta Math. 118(2), 135–149 (2005)\nLuo, W., Rudnick, Z., Sarnak, P.: On the generalized ramanujan conjecture for gl (n). In: Proceedings of Symposia in Pure Mathematics, vol. 66, pp. 301–310. American Mathematical Society, Providenc (1998, 1999)\nMurty, M.R.: Selberg’s conjectures and Artin \\(L\\)-functions. Bull. Am. Math. Soc. 31(1), 1–14 (1994)\nMurty, M.R.: Selberg’s conjectures and Artin \\(L\\)-functions. II. In: Current Trends in Mathematics and Physics, pp. 154–168. Narosa, New Delhi (1995)\nOdoni, R.W.K.: Solution of a generalised version of a problem of rankin on sums of powers of cusp-form coefficients. Acta Arith. 104(3), 201–223 (2002)\nRamakrishnan, D.: On the coefficients of cusp forms. Math. Res. Lett. 4(2–3), 295–307 (1997)\nRamakrishnan, D.: Modularity of the Rankin–Selberg \\(L\\)-series, and multiplicity one for \\({\\rm SL}(2)\\). Ann. Math. 152(1), 45–111 (2000)\nRankin, R.A.: Sums of powers of cusp form coefficients II. Math. Ann. 272(4), 593–600 (1985)\nRankin, R.A.: Sums of cusp form coefficients. In: Automorphic Forms and Analytic Number Theory (Montreal, PQ, 1989), pp. 115–121. Univ. Montréal, Montreal (1990)\nRudnick, Z., Sarnak, P.: Zeros of principal \\(L\\)-functions and random matrix theory. Duke Math. J. 81(2), 269–322. A celebration of John F. Nash, Jr. (1996)\nSalié, H.: Zur abschätzung der fourierkoeffizienten ganzer modulformen. Math. Zeitschrift 36(1), 263–278 (1933)\nSelberg, A.: Old and new conjectures and results about a class of Dirichlet series. In: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), pp. 367–385. Univ. Salerno, Salerno (1992)\nTang, H., Jie, W.: Fourier coefficients of symmetric power \\(L\\)-functions. J. Number Theory 167, 147–160 (2016)\nWalfisz, A.: über die koeffizientensummen einiger modulformen. Math. Ann. 108(1), 75–90 (1933)\nWeil, A.: On some exponential sums. Proc. Natl. Acad. Sci. USA 34(5), 204 (1948)\nWilton, J.R.: A note on ramanujan’s arithmetical function \\(\\tau \\)(n). In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 25, pp. 121–129. Cambridge University Press (1929)\nWu, J., Xu, Z.: Power sums of Hecke eigenvalues of maass cusp forms. Ramanujan J. 36(3), 439–453 (2015)\nWu, J., Ye, Y.: Hypothesis H and the prime number theorem for automorphic representations. Funct. Approx. Comment. Math. 37(2), 461–471 (2007)",{"VOID":106},"10.1007\u002Fs11139-020-00360-0","PUBLICATION","VERIFIED","2024-05-14T14:06:56.262+00:00","Auto Verify","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs11139-020-00360-0",[113],{"id":114,"sortIndex":19,"researcher":18,"roles":115,"affiliations":117,"properties":126,"displayName":128,"givenName":18,"familyName":18},"d66498be-d7d3-4cd8-9f87-061f230da78d",[116],"AUTHOR",[118],{"id":119,"sortIndex":19,"affiliation":120,"properties":18},"ab86d5c2-6ede-47fe-80a9-4956cd3e25b9",{"id":119,"createTime":18,"updateTime":18,"relativeEntities":121,"slug":18,"properties":122,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":125,"statistic":18},[],{"title":123},{"VI":124},"253-37 Caltech, Pasadena, USA",[],{"title":127,"gsAuthor":129},{"VI":128},"Liyang Yang",{"VOID":130},"[\"cQTdkzoAAAAJ\"]","ARTICLE",{"url":111,"publisher":133,"properties":174},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":134,"slug":10,"properties":135,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":138,"manageAffiliations":143,"indexDatabases":154,"url":18,"thumbnailPath":18,"statistic":169,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":136,"title":137},{"VOID":13},{"VOID":15},[139],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":140,"label":141,"description":142,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[144,149],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":145,"slug":18,"properties":146,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":148,"statistic":18},[],{"title":147},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":150,"slug":18,"properties":151,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":153,"statistic":18},[],{"title":152},{"EN":41},[],[155,162],{"id":45,"indexDatabase":156,"url":58,"indexYears":18,"academicFieldIds":161,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":157,"label":158,"description":159,"key":54,"publicationTags":160,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":163,"url":73,"indexYears":74,"academicFieldIds":168,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":164,"label":165,"description":166,"key":70,"publicationTags":167,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":170,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":171,"totalCitation":19,"totalCitationByYear":172,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":173,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":175,"volume":177},{"VOID":176},"203-234",{"VOID":178},"56",{"total":19,"publishYear":180,"statisticByYear":181},2021,{},"2021-02-18","ERROR_IN_ANALYZE_CITATION","2026-08-15T00:34:14.506+00:00",[56,77],false,{"id":188,"createTime":189,"updateTime":190,"relativeEntities":191,"slug":192,"properties":193,"entityType":107,"verifyStatus":108,"verifyTime":204,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":205,"fullTextUrl":18,"authors":206,"publicationType":131,"publisherRelationship":222,"citationCount":18,"citationInfo":18,"publishDate":269,"publishYear":270,"citationAnalyzeStatus":271,"lastCitationAnalyze":272,"indexDatabases":273,"openAccess":18,"references":18,"isForceReanalyzing":186},"7ae7a834-0622-4538-8172-314bcac02aa6","2024-01-27T09:38:01.754+00:00","2026-08-13T20:12:59.815+00:00",[],"A-Roth-type-theorem-with-mixed-powers",{"abstract":194,"title":196,"gsPaper":198,"references":200,"doi":202},{"EN":195},"Let \n$$c_1,c_2,c_3$$\n\n be nonzero integers such that \n$$c_1+c_2+c_3=0$$\n\n. We consider the mixed power equation \n$$c_1(p_1^2+p_1'^3)+c_2(p_2^2+p_2'^3)+c_3(p_3^2+p_3'^3)=0$$\n\n where \n$$p_1,p_2,p_3$$\n\n belong to a certain set \n$${\\mathcal {A}}$$\n\n of primes and \n$$p_1',p_2',p_3'$$\n\n belong to another set \n$${\\mathcal {A}}'$$\n\n of primes. We prove a Roth-type result that whenever the densities of \n$${\\mathcal {A}}$$\n\n and \n$${\\mathcal {A}}'$$\n\n satisfy a certain lower bound, then the above equation has nontrivial solutions. The same method can be generalized to deduce analogous results for other equations involving mixed powers of higher degrees.",{"EN":197},"A Roth-type theorem with mixed powers",{"VOID":199},"[]",{"VOID":201},"Bloom, T.F.: Translation invariant equations and the method of Sanders. Bull. Lond. Math. Soc. 44, 1050–1067 (2012)\nBloom, T.F.: A quantitative improvement for Roth’s theorem on arithmetic progressions. J. Lond. Math. Soc. 93, 643–663 (2016)\nBourgain, J.: On \\(\\varLambda (p)\\)-subsets of squares. Isr. J. Math. 67, 291–311 (1989)\nBourgain, J.: On triples in arithmetic progression. Geom. Funct. Anal. 9, 968–984 (1999)\nBourgain, J.: Roth’s theorem on progressions revisited. J. Anal. Math. 104, 155–192 (2008)\nBrowning, T.D., Prendiville, S.M.: A transference approach to a Roth-type theorem in the squares. Int. Math. Res. Not. 7, 2219–2248 (2017)\nChow, S.: Roth–Waring–Goldbach. Int. Math. Res. Not. 8, 2341–2374 (2018)\nGowers, W.T.: Decompositions, approximate structure, transference, and the Hahn–Banach theorem. Bull. Lond. Math. Soc. 42, 573–606 (2010)\nGreen, B.: Roth’s theorem in the primes. Ann. Math. 161, 1609–1636 (2005)\nGreen, B., Tao, T.: The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167, 481–547 (2008)\nHeath-Brown, D.R.: Integer sets containing no arithmetic progressions. J. Lond. Math. Soc. 35, 385–394 (1987)\nHua, L.K.: Additive Theory of Prime Numbers. Translations of Mathematical Monographs, vol. 13. American Mathematical Society, Providence (1965)\nPrendiville, S.M.: Four Variants of the Fourier-Analytic Transference Principle. arXiv:1509.09200 (2015)\nReingold, O., Trevisan, L., Tulsiani, M., Vadhan, S.: New Proofs of the Green–Tao–Ziegler Dense Model Theorem: An Exposition. arXiv:0806.0381 (2008)\nRoth, K.F.: On certain sets of integers. J. Lond. Math. Soc. 28, 104–109 (1953)\nSanders, T.: On Roth’s theorem on progressions. Ann. Math. 174, 619–636 (2011)\nSanders, T.: On certain other sets of integers. J. Anal. Math. 116, 53–82 (2012)\nSzemerédi, E.: On sets of integers containing no \\(k\\) elements in arithmetic progression. Acta Arith. 27, 299–345 (1975)",{"VOID":203},"10.1007\u002Fs11139-019-00148-x","2024-09-04T23:24:03.431+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs11139-019-00148-x",[207],{"id":208,"sortIndex":19,"researcher":18,"roles":209,"affiliations":210,"properties":219,"displayName":221,"givenName":18,"familyName":18},"c08397fa-ad23-440a-b976-4c000fb16ce5",[116],[211],{"id":212,"sortIndex":19,"affiliation":213,"properties":18},"0978018d-3086-4101-9299-7684a8656d2c",{"id":212,"createTime":18,"updateTime":18,"relativeEntities":214,"slug":18,"properties":215,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":218,"statistic":18},[],{"title":216},{"VI":217},"The University of Hong Kong, Hong Kong, Hong Kong",[],{"title":220},{"VI":221},"Tak Wing Ching",{"url":205,"publisher":223,"properties":264},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":224,"slug":10,"properties":225,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":228,"manageAffiliations":233,"indexDatabases":244,"url":18,"thumbnailPath":18,"statistic":259,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":226,"title":227},{"VOID":13},{"VOID":15},[229],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":230,"label":231,"description":232,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[234,239],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":235,"slug":18,"properties":236,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":238,"statistic":18},[],{"title":237},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":240,"slug":18,"properties":241,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":243,"statistic":18},[],{"title":242},{"EN":41},[],[245,252],{"id":45,"indexDatabase":246,"url":58,"indexYears":18,"academicFieldIds":251,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":247,"label":248,"description":249,"key":54,"publicationTags":250,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":253,"url":73,"indexYears":74,"academicFieldIds":258,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":254,"label":255,"description":256,"key":70,"publicationTags":257,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":260,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":261,"totalCitation":19,"totalCitationByYear":262,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":263,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":265,"volume":267},{"VOID":266},"581-604",{"VOID":268},"52","2019-07-03",2019,"ERROR_IN_GET_PLATFORM_ID","2026-08-13T20:12:59.814+00:00",[56,77],{"id":275,"createTime":276,"updateTime":277,"relativeEntities":278,"slug":279,"properties":280,"entityType":107,"verifyStatus":108,"verifyTime":291,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":292,"fullTextUrl":293,"authors":294,"publicationType":131,"publisherRelationship":327,"citationCount":18,"citationInfo":18,"publishDate":376,"publishYear":377,"citationAnalyzeStatus":17,"lastCitationAnalyze":277,"indexDatabases":378,"openAccess":18,"references":18,"isForceReanalyzing":186},"be5fd23c-1c81-42e8-8150-4cb93d9686f0","2024-01-27T19:22:23.444+00:00","2026-07-23T17:32:57.725+00:00",[],"Generators-of-Jacobi-forms-are-Poincar%C3%A9-series",{"abstract":281,"title":283,"gsPaper":285,"references":287,"doi":289},{"EN":282},"The ring of Jacobi forms of even weights is generated by the weak Jacobi forms $$\\phi _{-2,1}$$ and $$\\phi _{0,1}$$ . Bringmann and the first author expressed $$\\phi _{-2,1}$$ as a specialization of a Maass–Jacobi–Poincaré series. In this paper, we extend the domain of absolute convergence of Maass–Jacobi–Poincaré series which allows us to show that $$\\phi _{0,1}$$ is also a Poincaré series.",{"EN":284},"Generators of Jacobi forms are Poincaré series",{"VOID":286},"[\"15916575131725915762\"]",{"VOID":288},"Boylan, H., Skoruppa, N.-P., Zhou, H.: Arithmetic theory of skew-holomorphic Jacobi forms. Preprint (2016)\nBringmann, K.: Applications of Poincaré series on Jacobi groups. PhD thesis, University of Cologne, Germany (2004)\ncitation_journal_title=Trans. Am. Math. Soc.; citation_title=Harmonic Maass–Jacobi forms with singularities and a theta-like decomposition; citation_author=K Bringmann, M Raum, O Richter; citation_volume=367; citation_issue=9; citation_publication_date=2015; citation_pages=6647-6670; citation_doi=10.1090\u002FS0002-9947-2015-06418-6; citation_id=CR3\ncitation_journal_title=Adv. Math.; citation_title=Zagier-type dualites and lifting maps for harmonic Maass–Jacobi forms; citation_author=K Bringmann, O Richter; citation_volume=225; citation_issue=4; citation_publication_date=2010; citation_pages=2298-2315; citation_doi=10.1016\u002Fj.aim.2010.03.033; citation_id=CR4\ncitation_journal_title=Int. J. Number Theory; citation_title=Exact formulas for Fourier coefficients of Jacobi forms; citation_author=K Bringmann, O Richter; citation_volume=7; citation_issue=3; citation_publication_date=2011; citation_pages=825-833; citation_doi=10.1142\u002FS1793042111004617; citation_id=CR5\ncitation_journal_title=Int. Math. Res. Notices; citation_title=On Jacobi Poincaré series of small weight; citation_author=K Bringmann, T Yang; citation_volume=6; citation_publication_date=2007; citation_pages=2891-2912; citation_id=CR6\ncitation_title=The Theory of Jacobi Forms; citation_publication_date=1985; citation_id=CR7; citation_author=M Eichler; citation_author=D Zagier; citation_publisher=Birkhäuser\nNIST Digital Library of Mathematical Functions. \n                    http:\u002F\u002Fdlmf.nist.gov\u002F\n                    \n                  , Release 1.0.10 of 2015-08-07\nSkoruppa, N.-P.: Developments in the Theory of Jacobi Forms. Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, pp. 167–185 (1990)",{"VOID":290},"10.1007\u002Fs11139-016-9825-x","2024-06-25T14:32:40.343+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs11139-016-9825-x","https:\u002F\u002Flink.springer.com\u002Fcontent\u002Fpdf\u002F10.1007\u002Fs11139-016-9825-x.pdf",[295,312],{"id":296,"sortIndex":19,"researcher":18,"roles":297,"affiliations":298,"properties":307,"displayName":309,"givenName":18,"familyName":18},"331d1107-240e-4ec4-a244-dadbc456b1a2",[116],[299],{"id":300,"sortIndex":19,"affiliation":301,"properties":18},"b49e00ae-9324-45a5-9518-172cdd1f597e",{"id":300,"createTime":18,"updateTime":18,"relativeEntities":302,"slug":18,"properties":303,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":306,"statistic":18},[],{"title":304},{"VI":305},"Department of Mathematics, University of North Texas, Denton, USA",[],{"title":308,"gsAuthor":310},{"VI":309},"Richter, Olav K.",{"VOID":311},"[\"6Lwj9AUAAAAJ\"]",{"id":313,"sortIndex":82,"researcher":18,"roles":314,"affiliations":315,"properties":324,"displayName":326,"givenName":18,"familyName":18},"caa3ba54-d5a4-4c1c-8c83-96fb681581c1",[116],[316],{"id":317,"sortIndex":19,"affiliation":318,"properties":18},"21a48a89-4f07-4a3d-ac13-b94361eecd49",{"id":317,"createTime":18,"updateTime":18,"relativeEntities":319,"slug":18,"properties":320,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":323,"statistic":18},[],{"title":321},{"VI":322},"Department of Mathematics, State University of New York at Brockport, Brockport, USA",[],{"title":325},{"VI":326},"Skogman, Howard",{"url":292,"publisher":328,"properties":369},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":329,"slug":10,"properties":330,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":333,"manageAffiliations":338,"indexDatabases":349,"url":18,"thumbnailPath":18,"statistic":364,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":331,"title":332},{"VOID":13},{"VOID":15},[334],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":335,"label":336,"description":337,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[339,344],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":340,"slug":18,"properties":341,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":343,"statistic":18},[],{"title":342},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":345,"slug":18,"properties":346,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":348,"statistic":18},[],{"title":347},{"EN":41},[],[350,357],{"id":45,"indexDatabase":351,"url":58,"indexYears":18,"academicFieldIds":356,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":352,"label":353,"description":354,"key":54,"publicationTags":355,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":358,"url":73,"indexYears":74,"academicFieldIds":363,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":359,"label":360,"description":361,"key":70,"publicationTags":362,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":365,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":366,"totalCitation":19,"totalCitationByYear":367,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":368,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"issue":370,"pages":372,"volume":374},{"VOID":371},"3",{"VOID":373},"639-645",{"VOID":375},"45","2018-04-01",2018,[56,77],{"id":380,"createTime":381,"updateTime":382,"relativeEntities":383,"slug":384,"properties":385,"entityType":107,"verifyStatus":108,"verifyTime":394,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":395,"fullTextUrl":18,"authors":396,"publicationType":131,"publisherRelationship":412,"citationCount":19,"citationInfo":459,"publishDate":461,"publishYear":180,"citationAnalyzeStatus":17,"lastCitationAnalyze":382,"indexDatabases":462,"openAccess":18,"references":463,"isForceReanalyzing":186},"2c17539e-de16-4efa-8039-cfe63e3928a8","2023-11-27T15:40:46.086+00:00","2026-07-22T05:53:48.133+00:00",[],"A-supersingular-coincidence",{"abstract":386,"title":388,"gsPaper":390,"doi":392},{"EN":387},"The 15 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 are called the supersingular primes: they occur in several contexts in number theory and also, strikingly, they are the primes that divide the order of the Monster. It is also known that the moduli space of (1, p)-polarised abelian surfaces is of general type for these primes. In this note, we explain that apparently coincidental fact by relating it to other number-theoretic occurences of the supersingular primes.",{"EN":389},"A supersingular coincidence",{"VOID":391},"[\"9681973247370811349\"]",{"VOID":393},"10.1007\u002Fs11139-021-00526-4","2024-05-02T05:17:56.470+00:00","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs11139-021-00526-4",[397],{"id":398,"sortIndex":19,"researcher":18,"roles":399,"affiliations":400,"properties":409,"displayName":411,"givenName":18,"familyName":18},"377b2d51-cc0b-4564-bff7-2ee8edf5b6f1",[116],[401],{"id":402,"sortIndex":19,"affiliation":403,"properties":18},"3ec4f5d4-33a6-44d0-9a69-cfcd156acc30",{"id":402,"createTime":18,"updateTime":18,"relativeEntities":404,"slug":18,"properties":405,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":408,"statistic":18},[],{"title":406},{"VI":407},"Department of Mathematical Sciences, University of Bath, Bath, UK",[],{"title":410},{"VI":411},"G. K. Sankaran",{"url":395,"publisher":413,"properties":454},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":414,"slug":10,"properties":415,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":418,"manageAffiliations":423,"indexDatabases":434,"url":18,"thumbnailPath":18,"statistic":449,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":416,"title":417},{"VOID":13},{"VOID":15},[419],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":420,"label":421,"description":422,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[424,429],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":425,"slug":18,"properties":426,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":428,"statistic":18},[],{"title":427},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":430,"slug":18,"properties":431,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":433,"statistic":18},[],{"title":432},{"EN":41},[],[435,442],{"id":45,"indexDatabase":436,"url":58,"indexYears":18,"academicFieldIds":441,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":437,"label":438,"description":439,"key":54,"publicationTags":440,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":443,"url":73,"indexYears":74,"academicFieldIds":448,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":444,"label":445,"description":446,"key":70,"publicationTags":447,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":450,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":451,"totalCitation":19,"totalCitationByYear":452,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":453,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":455,"volume":457},{"VOID":456},"609-613",{"VOID":458},"59",{"total":19,"publishYear":180,"statisticByYear":460},{},"2021-12-20",[56,77],[464,470,476,479,482,485,488,491,494,497,500,504,507,510,513,516,519],{"id":465,"text":466,"url":467,"identifiers":468},"4c68646b-0035-4279-8000-0006b275d4fa","Aricheta, V.M.: Supersingular elliptic curves and moonshine. SIGMA Symmetry Integr. Geom. Methods Appl. 15, 17 (2019)","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10440-022-00541-7",{"doi":469},"10.1007\u002Fs10440-022-00541-7",{"id":471,"text":472,"url":473,"identifiers":474},"298e5e69-04bf-4f02-8952-b9be03f65609","Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405–444 (1992)","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF01232032",{"doi":475},"10.1007\u002Fbf01232032",{"id":465,"text":477,"url":467,"identifiers":478},"Duncan, J.F.R., Ono, K.: The Jack Daniels problem. J. Number Theory 161, 230–239 (2016)",{"doi":469},{"id":465,"text":480,"url":467,"identifiers":481},"Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Birkhäuser Boston, Inc., Boston (1985)",{"doi":469},{"id":465,"text":483,"url":467,"identifiers":484},"Erdenberger, C.: The Kodaira dimension of certain moduli spaces of Abelian surfaces. Math. Nachr. 274–275, 32–39 (2004)",{"doi":469},{"id":465,"text":486,"url":467,"identifiers":487},"Gritsenko, V.: Irrationality of the moduli spaces of polarized Abelian surfaces. Int. Math. Res. Not. 6, 235–243 (1994)",{"doi":469},{"id":465,"text":489,"url":467,"identifiers":490},"Gritsenko, V.: Irrationality of the moduli spaces of polarized abelian surfaces. Abelian varieties (Egloffstein, 1993), 63–84, de Gruyter, Berlin (1995)",{"doi":469},{"id":465,"text":492,"url":467,"identifiers":493},"Gritsenko, V., Sankaran, G.K.: Moduli of Abelian surfaces with a \\((1, p^2)\\) polarisation. Izv. Ross. Akad. Nauk. Ser. Mat. 66, 19–26 (1997)",{"doi":469},{"id":18,"text":495,"url":18,"identifiers":496},"Gross, M., Popescu, S.: The moduli space of \\((1,11)\\)-polarized Abelian surfaces is unirational. Compos. Math. 126, 1–23 (2001)",{},{"id":465,"text":498,"url":467,"identifiers":499},"Gross, M., Popescu, S.: Calabi-Yau three-folds and moduli of abelian surfaces II. Trans. Am. Math. Soc. 363, 3573–3599 (2011)",{"doi":469},{"id":18,"text":501,"url":18,"identifiers":502},"He, Y.-H., McKay, J.: Sporadic and exceptional. arXiv:1505.06742, (2015)",{"arxiv":503},"arXiv:1505.06742",{"id":18,"text":505,"url":18,"identifiers":506},"Hulek, K., Kahn, C., Weintraub, S.: Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions. Expositions in Mathematics, vol. 12. de Gruyter, Berlin (1993)",{},{"id":465,"text":508,"url":467,"identifiers":509},"Ogg, A.: Automorphismes de courbes modulaires. Séminaire Delange-Pisot Poitou (16e année: 1974\u002F75), Théorie des nombres, Fasc. 1, Exp. No. 7, 8 pp. Secrétariat Mathématique, Paris (1975)",{"doi":469},{"id":465,"text":511,"url":467,"identifiers":512},"O’Grady, K.: On the Kodaira dimension of moduli spaces of Abelian surfaces. Compos. Math. 72, 121–163 (1989)",{"doi":469},{"id":465,"text":514,"url":467,"identifiers":515},"Sankaran, G.K.: Moduli of polarised Abelian surfaces. Math. Nachr. 188, 321–340 (1997)",{"doi":469},{"id":465,"text":517,"url":467,"identifiers":518},"Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J. (1971)",{"doi":469},{"id":520,"text":521,"url":522,"identifiers":523},"beb7422a-7bbc-4137-bfe0-29c76816cd9e","Skoruppa, N., Zagier, D.: Jacobi forms and a certain space of modular forms. Invent. Math. 94, 113–146 (1988)","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF01394347",{"doi":524},"10.1007\u002Fbf01394347",{"id":526,"createTime":527,"updateTime":528,"relativeEntities":529,"slug":530,"properties":531,"entityType":107,"verifyStatus":108,"verifyTime":540,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":541,"fullTextUrl":18,"authors":542,"publicationType":131,"publisherRelationship":560,"citationCount":19,"citationInfo":607,"publishDate":610,"publishYear":608,"citationAnalyzeStatus":17,"lastCitationAnalyze":611,"indexDatabases":612,"openAccess":18,"references":613,"isForceReanalyzing":186},"b179cee5-53ba-44a5-801f-c2e1f534e56b","2024-01-18T02:52:14.470+00:00","2026-07-20T16:43:45.706+00:00",[],"Some-combinatorial-properties-of-hook-lengths-contents-and-parts-of-partitions",{"abstract":532,"title":534,"gsPaper":536,"doi":538},{"EN":533},"The main result of this paper is a generalization of a conjecture of Guoniu Han, originally inspired by an identity of Nekrasov and Okounkov. Our result states that if F is any symmetric function (say over ℚ) and if \n\n                  \n                    \n                  \n                  $$\\Phi_n(F)=\\frac{1}{n!}\\sum_{\\lambda\\vdash n}f_\\lambda^2F(h_u^2:u\\in\\lambda),$$\n                 \nwhere h\n                        \n                  u\n                 denotes the hook length of the square u of the partition λ of n and f\n                        \n                  λ\n                 is the number of standard Young tableaux of shape λ, then Φ\n                  n\n                (F) is a polynomial function of n. A similar result is obtained when F(h\n                        \n                  \n                    \n                  \n                  2\n                :u∈λ) is replaced with a function that is symmetric separately in the contents c\n                        \n                  u\n                 of λ and the shifted parts λ\n                        \n                  i\n                +n−i of λ.",{"EN":535},"Some combinatorial properties of hook lengths, contents, and parts of partitions",{"VOID":537},"[\"5889526155280104003\"]",{"VOID":539},"10.1007\u002Fs11139-009-9185-x","2024-04-23T19:11:20.619+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs11139-009-9185-x",[543],{"id":544,"sortIndex":19,"researcher":18,"roles":545,"affiliations":546,"properties":555,"displayName":557,"givenName":18,"familyName":18},"ab2994c1-0f74-4b42-a4a0-52886fe385a9",[116],[547],{"id":548,"sortIndex":19,"affiliation":549,"properties":18},"15504116-c418-4d76-a49f-dd429cc4a708",{"id":548,"createTime":18,"updateTime":18,"relativeEntities":550,"slug":18,"properties":551,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":554,"statistic":18},[],{"title":552},{"VI":553},"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA",[],{"title":556,"gsAuthor":558},{"VI":557},"Richard P. Stanley",{"VOID":559},"[\"Rc5HbpQAAAAJ\"]",{"url":541,"publisher":561,"properties":602},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":562,"slug":10,"properties":563,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":566,"manageAffiliations":571,"indexDatabases":582,"url":18,"thumbnailPath":18,"statistic":597,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":564,"title":565},{"VOID":13},{"VOID":15},[567],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":568,"label":569,"description":570,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[572,577],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":573,"slug":18,"properties":574,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":576,"statistic":18},[],{"title":575},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":578,"slug":18,"properties":579,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":581,"statistic":18},[],{"title":580},{"EN":41},[],[583,590],{"id":45,"indexDatabase":584,"url":58,"indexYears":18,"academicFieldIds":589,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":585,"label":586,"description":587,"key":54,"publicationTags":588,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":591,"url":73,"indexYears":74,"academicFieldIds":596,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":592,"label":593,"description":594,"key":70,"publicationTags":595,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":598,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":599,"totalCitation":19,"totalCitationByYear":600,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":601,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":603,"volume":605},{"VOID":604},"91-105",{"VOID":606},"23",{"total":19,"publishYear":608,"statisticByYear":609},2009,{},"2009-10-15","2026-07-20T16:43:45.705+00:00",[56,77],[614,618,621,624,627,630,633,636,640,643,646,649,652],{"id":18,"text":615,"url":18,"identifiers":616},"Amdeberhan, T.: Differential operators, shifted parts, and hook lengths. Preprint arXiv:0807.2473",{"arxiv":617},"arXiv:0807.2473",{"id":465,"text":619,"url":467,"identifiers":620},"Andrews, G., Goulden, I., Jackson, D.M.: Generalizations of Cauchy’s summation formula for Schur functions. Trans. Am. Math. Soc. 310, 805–820 (1988)",{"doi":469},{"id":18,"text":622,"url":18,"identifiers":623},"Frame, J.S., de Robinson, G.B., Thrall, R.M.: The hook graphs of S n . Can. J. Math. 6, 316–324 (1954)",{},{"id":465,"text":625,"url":467,"identifiers":626},"Fujii, S., Kanno, H., Moriyama, S., Okada, S.: Instanton calculus and chiral one-point functions in supersymmetric gauge theories. Adv. Theor. Math. Phys. 12, 1401–1428 (2008)",{"doi":469},{"id":465,"text":628,"url":467,"identifiers":629},"Hanlon, P.J., Stanley, R., Stembridge, J.R.: Some combinatorial aspects of the spectra of normally distributed random matrices. Contemp. Math. 158, 151–174 (1992)",{"doi":469},{"id":465,"text":631,"url":467,"identifiers":632},"Han, G.-N.: The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension, and applications. Preprint arXiv:0805.1398",{"doi":469},{"id":465,"text":634,"url":467,"identifiers":635},"Han, G.-N.: Some conjectures and open problems on partition hook lengths. Preprint available at www-irma.u-strasbg.fr\u002F~guoniu\u002Fhook",{"doi":469},{"id":18,"text":637,"url":18,"identifiers":638},"Han, G.-N.: Hook lengths and shifted parts of partitions. Preprint arXiv:0807.1801",{"arxiv":639},"arXiv:0807.1801",{"id":18,"text":641,"url":18,"identifiers":642},"Lascoux, A.: Symmetric Functions and Combinatorial Operators on Polynomials. CBMS Regional Conference Series in Mathematics, vol. 99. American Mathematical Society, Providence (2003)",{},{"id":18,"text":644,"url":18,"identifiers":645},"Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions, in the unity of mathematics. In: Progress in Mathematics, vol. 244, pp. 525–596. Birkhäuser Boston, Boston (2006)",{},{"id":18,"text":647,"url":18,"identifiers":648},"Okada, S.: Private communication. Dated 7 July (2008)",{},{"id":465,"text":650,"url":467,"identifiers":651},"Panova, G.: Proof of a conjecture of Okada. Preprint arXiv:0811.3463",{"doi":469},{"id":18,"text":653,"url":18,"identifiers":654},"Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, New York\u002FCambridge (1999)",{},{"id":656,"createTime":657,"updateTime":658,"relativeEntities":659,"slug":660,"properties":661,"entityType":107,"verifyStatus":108,"verifyTime":672,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":673,"fullTextUrl":18,"authors":674,"publicationType":131,"publisherRelationship":690,"citationCount":18,"citationInfo":18,"publishDate":737,"publishYear":738,"citationAnalyzeStatus":17,"lastCitationAnalyze":658,"indexDatabases":739,"openAccess":18,"references":18,"isForceReanalyzing":186},"7e3e6aec-d967-431f-8162-523f40a1ef4f","2024-01-15T10:32:24.951+00:00","2026-07-20T11:03:14.562+00:00",[],"Fourth-Order-Linearly-Recurrent-Wythoff-Pairs",{"abstract":662,"title":664,"gsPaper":666,"references":668,"doi":670},{"EN":663},"An examination of the sizable literature on Wythoff pairs, their generalizations, and Beatty sequences shows a considerable emphasis on connections with second order recurrence relations and quadratic irrationalities. However, it does not seem to have been previously noticed that a subsequence of the classical Wythoff pairs satisfies the irreducible fourth order linear recurrence \n                  \n                    \n                  \n                  \n$$C_{n + 4} = 10C_{n + 3} - 16C_{n + 2} + 5C_{n + 1} + C_n . $$\n\n                 Thus, while Wythoff pairs are ordinarily associated with the roots of x2 − x − 1 = 0, there is a subset thereof that is associated with the roots of \n                  \n                    \n                  \n                  \n$$ x^4 - 10x^3 + 16x^2 - 5x - 1 = 0. $$\n\n                \n",{"EN":665},"Fourth Order Linearly Recurrent Wythoff Pairs",{"VOID":667},"[\"1657901216202795635\"]",{"VOID":669},"E.R. Berlekamp, J.H. Conway, and R.K. Guy, Winning Ways, Academic Press, London, 1982.\nT.C. Brown, “Descriptions of the characteristic sequence of an irrational,” Canad. Math. Bull. 36 (1993), 15–21.\nF.R.K. Chung and R.L. Graham, “On irregularities of distribution of real sequences,” Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 4001.\nA.S. Fraenkel, “Systems of numeration,” Amer. Math. Monthly 92 (1985), 105–114.\nA.S. Fraenkel, “Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups,” Trans. Amer. Math. Soc. 341 (1994), 639–664.\nA.S. Fraenkel, M. Mushkin, and U. Tassa, “Determination of [nθ] by its sequence of differences,” Canadian Math. Bull. 21 (1978), 441–446.\nA.S. Fraenkel, H. Porta, and K.B. Stolarsky, “Certain arithmetic semigroups,” in Proceedings of an International Conference on Analytic Number Theory, Progress in Mathematics, Birkhäuser, Boston, pp. 255–264, 1990.\nA.S. Fraenkel, H. Porta, and K.B. Stolarsky, “The almost PV behavior of some far from PV algebraic integers,” Discrete Math. 135 (1994), 93–101.\nJ.C. Lagarias, H. Porta, and K.B. Stolarsky, “Asymmetric tent map expansions I: Eventually periodic points,” J. London Math. Soc. 47(2), (1993), 542–556.\nJ.C. Lagarias, H. Porta, and K.B. Stolarsky, “Asymmetric tent map expansions II: Purely periodic points and preperiods of zero,” Illinois J. of Math. 38 (1994), 574–588.\nH. Porta and K.B. Stolarsky, “The edge of a golden semigroup,” Colloq, Math. Soc. János Bolyai (Number Theory) 51 (1987), 465–471.\nH. Porta and K.B. Stolarsky, “Half-silvered mirrors and Wythoff's game,” Canad. Math. Bull. 33(1), (1990), 119–125.\nH. Porta and K.B. Stolarsky, “Wythoff pairs as semigroup invariants,” Advances in Mathematics 85 (1991), 69–82.\nK.B. Stolarsky, “Beatty sequences, continued fractions, and certain shift operators,” Canad. Math. Bull. 19 (1976), 472–482.\nK.B. Stolarsky, “From Wythoff's nim to Chebyshev's inequality,” Amer. Math. Monthly 98 (1991), 889–900.\nK.B. Stolarsky, “Certain sequences of Wythoffian matrices and maximal geometric progressions therein,” J. Combin. Theory Ser A 68 (1994), 361–371.\nK.B. Stolarsky, “Positive factors of Wythoff matrices,” Semigroup Forum 53 (1996), 271–277.\nW.A. Wythoff, “A modification of the game of Nim,” Nieuw. Archief voor Viskunde 7(2), (1907), 199–200.",{"VOID":671},"10.1023\u002FA:1009776709280","2024-06-26T16:30:36.088+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1023\u002FA:1009776709280",[675],{"id":676,"sortIndex":19,"researcher":18,"roles":677,"affiliations":678,"properties":687,"displayName":689,"givenName":18,"familyName":18},"35321019-3c5d-412e-86f4-be7d9b3882ef",[116],[679],{"id":680,"sortIndex":19,"affiliation":681,"properties":18},"03b8e24a-619a-422a-963e-b53bae3b0b3f",{"id":680,"createTime":18,"updateTime":18,"relativeEntities":682,"slug":18,"properties":683,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":686,"statistic":18},[],{"title":684},{"VI":685},"Department of Mathematics, University of Illinois, Urbana",[],{"title":688},{"VI":689},"Kenneth B. Stolarsky",{"url":673,"publisher":691,"properties":732},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":692,"slug":10,"properties":693,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":696,"manageAffiliations":701,"indexDatabases":712,"url":18,"thumbnailPath":18,"statistic":727,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":694,"title":695},{"VOID":13},{"VOID":15},[697],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":698,"label":699,"description":700,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[702,707],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":703,"slug":18,"properties":704,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":706,"statistic":18},[],{"title":705},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":708,"slug":18,"properties":709,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":711,"statistic":18},[],{"title":710},{"EN":41},[],[713,720],{"id":45,"indexDatabase":714,"url":58,"indexYears":18,"academicFieldIds":719,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":715,"label":716,"description":717,"key":54,"publicationTags":718,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":721,"url":73,"indexYears":74,"academicFieldIds":726,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":722,"label":723,"description":724,"key":70,"publicationTags":725,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":728,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":729,"totalCitation":19,"totalCitationByYear":730,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":731,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":733,"volume":735},{"VOID":734},"441-448",{"VOID":736},"2","1998-12-01",1998,[56,77],{"id":741,"createTime":742,"updateTime":743,"relativeEntities":744,"slug":745,"properties":746,"entityType":107,"verifyStatus":108,"verifyTime":757,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":758,"fullTextUrl":18,"authors":759,"publicationType":131,"publisherRelationship":790,"citationCount":836,"citationInfo":837,"publishDate":840,"publishYear":838,"citationAnalyzeStatus":17,"lastCitationAnalyze":841,"indexDatabases":842,"openAccess":18,"references":18,"isForceReanalyzing":186},"c83822c1-f83f-43f4-80a2-03aa6ffbfe25","2023-12-27T15:41:29.807+00:00","2026-07-16T17:56:13.390+00:00",[],"A-mathrm-GL-3-analog-of-Selberg-s-result-on-S-t-",{"abstract":747,"title":749,"gsPaper":751,"references":753,"doi":755},{"EN":748},"Let \n                \n                  \n                \n                $$S(t,F):=\\pi ^{-1}\\arg L\\big (\\frac{1}{2}+it,F\\big ),$$\n                \n               where F is a Hecke–Maass cusp form for \n                \n                  \n                \n                $$\\mathrm {SL}_3({\\mathbb {Z}}).$$\n                \n               We establish an asymptotic formula for the spectral moments of S(t, F), and obtain several other results on S(t, F).",{"EN":750},"A $$\\mathrm {GL}_3$$ analog of Selberg’s result on S(t)",{"VOID":752},"[\"4068748496769311577\"]",{"VOID":754},"Backlund, R.J.: Sur las zéros de la fonction \\(\\zeta (s)\\) de Riemann. C. R. Acad. Sci. Paris 158, 1979–1981 (1914)\nBacklund, R.J.: Über die Nullstellen der Riemannschen Zetafunktion. Acta Math. 41, 345–375 (1918)\nBillingsley, P.: Probability and Measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics, Wiley, New York (1995)\nButtcane, J., Zhou, F.: Plancherel distribution of satake parameters of Maass Cusp forms on \\({{\\rm GL}}_3\\). International Mathematics Research Notices, to appear\nGoldfeld, D.: Automorphic Forms and \\(L\\)-Functions for the Group \\({{\\rm GL}} (n,{\\mathbb{R}})\\), with an appendix by Kevin A. Broughan, vol. 99. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2006)\nHejhal, D., Luo, W.: On a spectral analog of Selberg’s result on \\(S(T)\\). Int. Math. Res. Notices 3, 135–151 (1997)\nKim, H.H.: Functoriality for the exterior square of \\({{\\rm GL}}_4\\) and the symmetric fourth of \\({{\\rm GL}}_2\\). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. J. Am. Math. Soc. 16(1), 139–183 (2003)\nLittlewood, J.E.: On the zeros of the Riemann zeta-function. Proc. Camb. Philos. Soc. 2(24), 295–318 (1924)\nMacDonald, I.G.: Symmetric Functions and Hall Polynomials. With Contributions by A. Zelevinsky, 2nd edn. Oxford Science Publications. The Clarendon Press, Oxford Mathematical Monographs. Oxford University Press, New York (1995)\nvon Mangoldt, H.: Zu Riemanns Abhandlung “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”. J. Reine Angew. Math. 114, 255–305 (1895)\nvon Mangoldt, H.: Zur Verteilung der Nullstellen der Riemannschen Funktion \\(\\xi (t)\\). Math. Ann. 60(1), 1–19 (1905)\nSelberg, A.: On the Remainder in the Formula for \\(N(T)\\), the Number of Zeros of \\(\\zeta (s)\\) in the strip \\(0\u003Ct\u003CT\\). Avh. Norske Vid. Akad. Oslo. I.,. no. 1, p. 27; Collected Papers I, vol. 1989, pp. 179–203. Springer, Berlin (1944)\nSelberg, A.: Contributions to the theory of the Riemann zeta-function. Arch. Math. Naturvid. 48(5), 89–155; Collected Papers I, vol. 1989, pp. 214–280. Springer, Berlin (1946)\nSelberg, A.: Contributions to the theory of Dirichlet’s \\(L\\)-functions. Skr. Norske Vid. Akad. Oslo. I.,: (1946). no. 3, p. 62; Collected Papers I, vol. 1989, pp. 281–340. Springer, Berlin (1946)\nTitchmarsh, E.C.: The zeros of the Riemann zeta-function. Proc. R. Soc. 151, 234–255 (1935)",{"VOID":756},"10.1007\u002Fs11139-020-00308-4","2024-05-07T15:25:54.878+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs11139-020-00308-4",[760,775],{"id":761,"sortIndex":19,"researcher":18,"roles":762,"affiliations":763,"properties":772,"displayName":774,"givenName":18,"familyName":18},"eb71cc10-18b4-43f0-8da8-02469a9fb8f8",[116],[764],{"id":765,"sortIndex":19,"affiliation":766,"properties":18},"27a7a9ea-44bc-465f-a5b2-43f4bb4e72a1",{"id":765,"createTime":18,"updateTime":18,"relativeEntities":767,"slug":18,"properties":768,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":771,"statistic":18},[],{"title":769},{"VI":770},"Department of Mathematics and Statistics, Washington State University, Pullman, USA",[],{"title":773},{"VI":774},"Sheng-Chi Liu",{"id":776,"sortIndex":82,"researcher":18,"roles":777,"affiliations":778,"properties":787,"displayName":789,"givenName":18,"familyName":18},"145ab623-d1a9-4f19-a735-1055d11f6efb",[116],[779],{"id":780,"sortIndex":19,"affiliation":781,"properties":18},"8e72c28f-3a09-4c3d-8f46-febcf79f5a08",{"id":780,"createTime":18,"updateTime":18,"relativeEntities":782,"slug":18,"properties":783,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":786,"statistic":18},[],{"title":784},{"VI":785},"Department of Mathematics and Statistics, University of Maine, Orono, USA",[],{"title":788},{"VI":789},"Shenhui Liu",{"url":758,"publisher":791,"properties":832},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":792,"slug":10,"properties":793,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":796,"manageAffiliations":801,"indexDatabases":812,"url":18,"thumbnailPath":18,"statistic":827,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":794,"title":795},{"VOID":13},{"VOID":15},[797],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":798,"label":799,"description":800,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[802,807],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":803,"slug":18,"properties":804,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":806,"statistic":18},[],{"title":805},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":808,"slug":18,"properties":809,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":811,"statistic":18},[],{"title":810},{"EN":41},[],[813,820],{"id":45,"indexDatabase":814,"url":58,"indexYears":18,"academicFieldIds":819,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":815,"label":816,"description":817,"key":54,"publicationTags":818,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":821,"url":73,"indexYears":74,"academicFieldIds":826,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":822,"label":823,"description":824,"key":70,"publicationTags":825,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":828,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":829,"totalCitation":19,"totalCitationByYear":830,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":831,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":833,"volume":835},{"VOID":834},"163-181",{"VOID":178},8,{"total":836,"publishYear":838,"statisticByYear":839},2020,{"2022":80,"2024":80,"2025":82,"2026":80},"2020-07-28","2026-07-16T17:56:13.389+00:00",[56,77],{"id":844,"createTime":845,"updateTime":846,"relativeEntities":847,"slug":848,"properties":849,"entityType":107,"verifyStatus":108,"verifyTime":860,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":861,"fullTextUrl":18,"authors":862,"publicationType":131,"publisherRelationship":880,"citationCount":80,"citationInfo":927,"publishDate":929,"publishYear":838,"citationAnalyzeStatus":17,"lastCitationAnalyze":930,"indexDatabases":931,"openAccess":18,"references":18,"isForceReanalyzing":186},"45d9d666-be18-46a4-8126-baae5ba5d5d2","2024-02-17T02:17:02.277+00:00","2026-07-15T12:45:39.152+00:00",[],"On-the-signs-of-Fourier-coefficients-of-Hilbert-cusp-forms",{"abstract":850,"title":852,"gsPaper":854,"references":856,"doi":858},{"EN":851},"We prove that given any \n$$\\epsilon > 0$$\n\n and a primitive adelic Hilbert cusp form f of weight \n$$k=(k_1,k_2,\\ldots ,k_n) \\in (2 {\\mathbb {Z}})^n$$\n\n and full level, there exists an integral ideal \n$${\\mathfrak {m}}$$\n\n with \n$$N({\\mathfrak {m}}) \\ll _{\\epsilon } Q_{f}^{9\u002F20+ \\epsilon } $$\n\n such that the \n$${\\mathfrak {m}}$$\n\n-th Fourier coefficient of \n$$C_{f} ({\\mathfrak {m}})$$\n\n of f is negative. Here n is the degree of the associated number field, \n$$N({\\mathfrak {m}})$$\n\n is the norm of integral ideal \n$${\\mathfrak {m}}$$\n\n and \n$$Q_{f}$$\n\n is the analytic conductor of f. In the case of arbitrary weights, we show that there is an integral ideal \n$${\\mathfrak {m}}$$\n\n with \n$$N({\\mathfrak {m}}) \\ll _{\\epsilon } Q_{f}^{1\u002F2 + \\epsilon }$$\n\n such that \n$$C_{f}({\\mathfrak {m}}) \u003C0$$\n\n. We also prove that when \n$$k=(k_1,k_2,\\ldots ,k_n) \\in (2 {\\mathbb {Z}})^n$$\n\n, asymptotically half of the Fourier coefficients are positive while half are negative.",{"EN":853},"On the signs of Fourier coefficients of Hilbert cusp forms",{"VOID":855},"[\"18401979852772045531\"]",{"VOID":857},"Barnet-Lamb, T., Gee, T., Geraghty, D.: The Sato–Tate conjecture for Hilbert modular forms. J. Am. Math. Soc. 24, 411–469 (2011)\nBlasius, D.: Hilbert Modular Forms and the Ramanujan Conjecture. Noncommutative Geometry and Number Theory, pp. 35–56. Springer, New York (2006)\nIvić, A.: On squarefree numbers with restricted prime factors. Stud. Sci. Math. Hung. 20, 189–192 (1985)\nIwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathemaical Society, Colloquium Publications, Providence (2004)\nKowalski, E., Lau, Y.K., Soundararajan, K., Wu, J.: On modular signs. Math. Proc. Camb. Philos. Soc. 149, 389–411 (2010)\nLebacque, P.: Generalized Merten’s and Brauer–Siegel theorems. Acta Arith. 130, 333–350 (2007)\nLucht, L.G., Reifenrath, K.: Mean value theorems in arithmetic semigroups. Acta Math. Hung. 93(1–2), 27–57 (2001)\nMatomäki, K.: On signs of Fourier coefficients of cusp forms. Math. Proc. Camb. Philos. Soc. 152, 207–222 (2012)\nMatomäki, K., Radziwiłł, M.: Sign changes of Hecke eigenvalues. Geom. Funct. Anal. 25, 1937–1955 (2015)\nMeher, J., Tanabe, N.: Sign changes of Fourier coefficients of Hilbert modular forms. J. Number Theory. 145, 230–244 (2014)\nMoree, P.: An interval result for the number field \\(\\psi (x, y)\\) function. Manuscr. Math. 76, 437–450 (1992)\nMurty, M.R.: Problems in Algebraic Number Theory, 2nd edn. Springer, New York (2005)\nQu, Y.: Linnik type problems for automorphic L-functons. J. Numb. Theory 130, 786–802 (2010)\nRaghuram, A., Tanabe, N.: Notes on the arithmetic of Hilbert modular forms. J. Ramanujan Math. Soc. 26, 261–319 (2011)\nSerre, J.P.: Quelques applications du theoreme de densite de Chebotarev. Publications mathematiques de l’I.H.E.S., tome 54, 123–201 (1981)\nShimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45, 637–679 (1978)\nTenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics 46. Cambridge University Press, Cambridge (1995)",{"VOID":859},"10.1007\u002Fs11139-019-00206-4","2024-05-17T12:12:59.082+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs11139-019-00206-4",[863],{"id":864,"sortIndex":19,"researcher":18,"roles":865,"affiliations":866,"properties":875,"displayName":877,"givenName":18,"familyName":18},"3791600b-84a2-4d9b-9c2c-93386359f641",[116],[867],{"id":868,"sortIndex":19,"affiliation":869,"properties":18},"a392a323-daee-42e2-8fe7-2931828440ec",{"id":868,"createTime":18,"updateTime":18,"relativeEntities":870,"slug":18,"properties":871,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":874,"statistic":18},[],{"title":872},{"VI":873},"Department of Mathematics, Indian Institute of Science, Bangalore, India",[],{"title":876,"gsAuthor":878},{"VI":877},"Ritwik Pal",{"VOID":879},"[\"5IewCigAAAAJ\"]",{"url":861,"publisher":881,"properties":922},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":882,"slug":10,"properties":883,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":886,"manageAffiliations":891,"indexDatabases":902,"url":18,"thumbnailPath":18,"statistic":917,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":884,"title":885},{"VOID":13},{"VOID":15},[887],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":888,"label":889,"description":890,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[892,897],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":893,"slug":18,"properties":894,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":896,"statistic":18},[],{"title":895},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":898,"slug":18,"properties":899,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":901,"statistic":18},[],{"title":900},{"EN":41},[],[903,910],{"id":45,"indexDatabase":904,"url":58,"indexYears":18,"academicFieldIds":909,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":905,"label":906,"description":907,"key":54,"publicationTags":908,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":911,"url":73,"indexYears":74,"academicFieldIds":916,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":912,"label":913,"description":914,"key":70,"publicationTags":915,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":918,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":919,"totalCitation":19,"totalCitationByYear":920,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":921,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":923,"volume":925},{"VOID":924},"467-481",{"VOID":926},"53",{"total":80,"publishYear":838,"statisticByYear":928},{"2021":82,"2022":82},"2020-01-22","2026-07-15T12:45:39.151+00:00",[56,77],{"id":933,"createTime":934,"updateTime":935,"relativeEntities":936,"slug":937,"properties":938,"entityType":107,"verifyStatus":108,"verifyTime":947,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":948,"fullTextUrl":18,"authors":949,"publicationType":131,"publisherRelationship":982,"citationCount":18,"citationInfo":18,"publishDate":1028,"publishYear":838,"citationAnalyzeStatus":1029,"lastCitationAnalyze":1030,"indexDatabases":1031,"openAccess":18,"references":1032,"isForceReanalyzing":186},"edc52df4-c774-49eb-8077-1cac8cc5c01c","2023-12-14T21:33:06.068+00:00","2026-07-15T03:47:21.527+00:00",[],"Representations-of-finite-number-of-quadratic-forms-with-same-rank",{"abstract":939,"title":941,"gsPaper":943,"doi":945},{"EN":940},"Let m, n be positive integers with \n                \n                  \n                \n                $$m\\le n$$\n                \n              . Let \n                \n                  \n                \n                $$\\kappa (m,n)$$\n                \n               be the largest integer k such that for any (positive definite and integral) quadratic forms \n                \n                  \n                \n                $$f_1,\\ldots ,f_k$$\n                \n               of rank m, there exists a quadratic form of rank n that represents \n                \n                  \n                \n                $$f_i$$\n                \n               for any i with \n                \n                  \n                \n                $$1\\le i \\le k$$\n                \n              . In this article, we determine the number \n                \n                  \n                \n                $$\\kappa (m,n)$$\n                \n               for any integer m with \n                \n                  \n                \n                $$1\\le m\\le 8$$\n                \n              , except for the cases when \n                \n                  \n                \n                $$(m,n)=(3,5)$$\n                \n               and (4, 6). In the exceptional cases, it will be proved that \n                \n                  \n                \n                $$1\\le \\kappa (3,5), \\kappa (4,6)\\le 2$$\n                \n              . We also discuss some related topics.",{"EN":942},"Representations of finite number of quadratic forms with same rank",{"VOID":944},"[\"15798762133573766152\"]",{"VOID":946},"10.1007\u002Fs11139-020-00314-6","2024-05-03T10:56:44.774+00:00","https:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs11139-020-00314-6",[950,965],{"id":951,"sortIndex":19,"researcher":18,"roles":952,"affiliations":953,"properties":962,"displayName":964,"givenName":18,"familyName":18},"7f04908f-04bd-49af-9458-c11393f44fed",[116],[954],{"id":955,"sortIndex":19,"affiliation":956,"properties":18},"27301efd-f532-463f-aecc-d1f30260c0ac",{"id":955,"createTime":18,"updateTime":18,"relativeEntities":957,"slug":18,"properties":958,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":961,"statistic":18},[],{"title":959},{"VI":960},"Research Institute of Mathematics, Seoul National University, Seoul, Korea",[],{"title":963},{"VI":964},"Daejun Kim",{"id":966,"sortIndex":82,"researcher":18,"roles":967,"affiliations":968,"properties":977,"displayName":979,"givenName":18,"familyName":18},"35be1e9d-6e47-459d-8ceb-a6a5a22e6f4e",[116],[969],{"id":970,"sortIndex":19,"affiliation":971,"properties":18},"00d90fef-4ae3-4679-a788-327206bb2697",{"id":970,"createTime":18,"updateTime":18,"relativeEntities":972,"slug":18,"properties":973,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":976,"statistic":18},[],{"title":974},{"VI":975},"Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, Korea",[],{"title":978,"gsAuthor":980},{"VI":979},"Byeong-Kweon Oh",{"VOID":981},"[\"5pKsAmsAAAAJ\"]",{"url":948,"publisher":983,"properties":1024},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":984,"slug":10,"properties":985,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":988,"manageAffiliations":993,"indexDatabases":1004,"url":18,"thumbnailPath":18,"statistic":1019,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":986,"title":987},{"VOID":13},{"VOID":15},[989],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":990,"label":991,"description":992,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[994,999],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":995,"slug":18,"properties":996,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":998,"statistic":18},[],{"title":997},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":1000,"slug":18,"properties":1001,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1003,"statistic":18},[],{"title":1002},{"EN":41},[],[1005,1012],{"id":45,"indexDatabase":1006,"url":58,"indexYears":18,"academicFieldIds":1011,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":1007,"label":1008,"description":1009,"key":54,"publicationTags":1010,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":1013,"url":73,"indexYears":74,"academicFieldIds":1018,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":1014,"label":1015,"description":1016,"key":70,"publicationTags":1017,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":1020,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":1021,"totalCitation":19,"totalCitationByYear":1022,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":1023,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":1025,"volume":1027},{"VOID":1026},"631-644",{"VOID":178},"2020-10-27","DONE_ANALYZE_CITATION","2026-07-15T03:47:21.526+00:00",[56,77],[1033,1036,1039,1042,1045,1048,1054,1057,1060,1063,1066,1069,1075,1078],{"id":465,"text":1034,"url":467,"identifiers":1035},"Bhargava, M.: On the Conway–Schneeberger fifteen theorem. Contemp. Math. 272, 27–37 (2000)",{"doi":469},{"id":18,"text":1037,"url":18,"identifiers":1038},"Cassels, J.K.S.: Rational Quadratic Forms. Academic Press, London (1978)",{},{"id":465,"text":1040,"url":467,"identifiers":1041},"Conway, J.H.: Universal quadratic forms and the Fifteen Theorem. Contemp. Math. 272, 23–26 (2000)",{"doi":469},{"id":18,"text":1043,"url":18,"identifiers":1044},"Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 290. Springer, New York (1999)",{},{"id":465,"text":1046,"url":467,"identifiers":1047},"Görtz, U.: Arithmetic intersection numbers. Astérisque 312, 15–24 (2007)",{"doi":469},{"id":1049,"text":1050,"url":1051,"identifiers":1052},"39992736-d2b8-48a8-b0af-abc9e2e8ddac","Gross, B.H., Keating, K.: On the intersection of modular correspondences. Invent. Math. 112, 225–245 (1993)","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01232433",{"doi":1053},"10.1007\u002FBF01232433",{"id":465,"text":1055,"url":467,"identifiers":1056},"Kim, B.M., Kim, M.-H., Oh, B.-K.: 2-Universal positive definite integral quinary quadratic forms. Contemp. Math. 249, 51–62 (1999)",{"doi":469},{"id":465,"text":1058,"url":467,"identifiers":1059},"Kitaoka, Y.: Arithmetic of Quadratic Forms. Cambridge University Press, Cambridge (1993)",{"doi":469},{"id":465,"text":1061,"url":467,"identifiers":1062},"Lee, I., Oh, B.-K., Yu, H.: A finiteness theorem for positive definite almost \\(n\\)-regular quadratic forms. J. Ramanujan Math. Soc. 35(1), 81–94 (2020)",{"doi":469},{"id":465,"text":1064,"url":467,"identifiers":1065},"Oh, B.-K.: Universal \\({\\mathbb{Z}}\\)-lattices of minimal rank. Proc. Am. Math. Soc. 128, 683–689 (1999)",{"doi":469},{"id":465,"text":1067,"url":467,"identifiers":1068},"O’Meara, O.T.: Introduction to Quadratic Forms. Springer, New York (1963)",{"doi":469},{"id":1070,"text":1071,"url":1072,"identifiers":1073},"61aed37b-4694-4d93-bd08-9533f8a24657","Ono, K., Soundararajan, K.: Ramanujan’s ternary quadratic form. Invent. Math. 130, 127–159 (1997)","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs002220050191",{"doi":1074},"10.1007\u002Fs002220050191",{"id":18,"text":1076,"url":18,"identifiers":1077},"Ramanujan, S.: On the expression of a number in the form \\(ax^2+by^2+cz^2+dw^2\\). Proc. Cambridge Phil. Soc. 19, 11–21 (1917)",{},{"id":1079,"text":1080,"url":1081,"identifiers":1082},"110a6013-bf99-4cea-b289-48683282b883","van der Waerden, B.L.: Die Reduktionstheorie der positiven quadratischen Formen. Acta Math. 96, 265–309 (1956)","https:\u002F\u002Fprojecteuclid.org\u002Fjournals\u002Facta-mathematica\u002Fvolume-96\u002Fissue-none\u002FDie-Reduktionstheorie-Der-Positiven-Quadratischen-Formen\u002F10.1007\u002FBF02392364.full",{"doi":1083},"10.1007\u002FBF02392364",{"id":1085,"createTime":1086,"updateTime":1087,"relativeEntities":1088,"slug":1089,"properties":1090,"entityType":107,"verifyStatus":108,"verifyTime":1100,"verifyNote":110,"languages":18,"translateLanguages":18,"viewCount":19,"primaryUrl":1101,"fullTextUrl":18,"authors":1102,"publicationType":131,"publisherRelationship":1133,"citationCount":18,"citationInfo":18,"publishDate":1180,"publishYear":1181,"citationAnalyzeStatus":271,"lastCitationAnalyze":1087,"indexDatabases":1182,"openAccess":18,"references":18,"isForceReanalyzing":186},"aa5f117f-77ec-459e-812f-22f91f0d45db","2024-01-15T05:58:50.388+00:00","2026-06-03T07:40:30.463+00:00",[],"On-the-smallest-number-with-a-given-number-of-divisors",{"abstract":1091,"title":1093,"gsPaper":1095,"references":1096,"doi":1098},{"EN":1092},"For a natural number \n                  \n                    \n                  \n                  $$n$$\n                  \n                    \n                  \n                , let \n                  \n                    \n                  \n                  $$A(n)$$\n                  \n                    \n                  \n                 denote the smallest natural number that has exactly \n                  \n                    \n                  \n                  $$n$$\n                  \n                    \n                  \n                 divisors. Let \n                  \n                    \n                  \n                  $$p_{1}p_{2}\\ldots p_{k}$$\n                  \n                    \n                  \n                 be the prime decomposition of \n                  \n                    \n                  \n                  $$n$$\n                  \n                    \n                  \n                 where the primes are given in decreasing order and the factors are not necessarily distinct. If \n                  \n                    \n                  \n                  $$q_{1},\\ldots ,q_{k}$$\n                  \n                    \n                  \n                 denote the first \n                  \n                    \n                  \n                  $$k$$\n                  \n                    \n                  \n                 primes and \n                  \n                    \n                  \n                  $$A(n)=q_{1}^{p_{1}-1}\\ldots q_{k}^{p_{k}-1}$$\n                  \n                    \n                  \n                , we say that \n                  \n                    \n                  \n                  $$n$$\n                  \n                    \n                  \n                 is ordinary. If \n                  \n                    \n                  \n                  $$n$$\n                  \n                    \n                  \n                 is not ordinary, we say it is extraordinary. In Brown (2006), it was shown that almost all numbers are ordinary and if \n                  \n                    \n                  \n                  $$E_{x}$$\n                  \n                    \n                  \n                 denotes the set of extraordinary numbers less than or equal to a positive real number \n                  \n                    \n                  \n                  $$x$$\n                  \n                    \n                  \n                , \n                  \n                    \n                  \n                  $$|E_{x}|=o\\left( \\frac{x}{2^{(\\log (\\log x))^{\\delta }}}\\right) $$\n                  \n                    \n                  \n                 for any \n                  \n                    \n                  \n                  $$\\displaystyle 0\u003C\\delta \u003C1\u002F2$$\n                  \n                    \n                  \n                . In what follows, we will prove that \n                  \n                    \n                  \n                  $$\\displaystyle |E_x|=\\frac{x}{\\log x}e^{\\Psi (x)}$$\n                  \n                    \n                  \n                 where \n                  \n                    \n                  \n                  $$\\displaystyle \\Psi (x)\\sim \\frac{1}{2\\log 2}\\frac{\\log _{(3)}^2x}{\\log _{(4)}x}$$\n                  \n                    \n                  \n                 and \n                  \n                    \n                  \n                  $$\\displaystyle \\log _{(r)}x$$\n                  \n                    \n                  \n                 denotes the \n                  \n                    \n                  \n                  $$r$$\n                  \n                    \n                  \n                -times iterated natural logarithm. We make use of the Prime Number Theorem as well as estimates of the size of sets where the number of total prime factors or distinct prime factors is fixed.",{"EN":1094},"On the smallest number with a given number of divisors",{"VOID":199},{"VOID":1097},"Brown, R.: The minimal number with a given number of divisors. J. Number Theory 116(1), 150–158 (2006)\nDusart, P.: The \\(k\\)th prime is greater than \\(k(\\ln k+\\ln \\ln k-1)\\) for \\(k\\ge 2\\). Math. Comput. 68, 411–415 (1999)\nGrost, M.E.: The smallest number with a given number of divisors. Am. Math. Mon. 75, 725–729 (1968)\nHardy, G.H., Ramanujan, S.: The normal number of prime factors of a number. Quart. J. Pure Appl. Math. 48, 76–92 (1920)\nMontgomery, H.L., Vaughan, R.C.: Multiplicative Number Theory I: Classical Theory. Cambridge University Press, Cambridge (2007)",{"VOID":1099},"10.1007\u002Fs11139-014-9572-9","2024-06-23T03:17:13.914+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs11139-014-9572-9",[1103,1118],{"id":1104,"sortIndex":19,"researcher":18,"roles":1105,"affiliations":1106,"properties":1115,"displayName":1117,"givenName":18,"familyName":18},"4eb0f149-7089-4933-a590-4ae3c10fc678",[116],[1107],{"id":1108,"sortIndex":19,"affiliation":1109,"properties":18},"4ca16f79-2583-4f02-a407-209096af8679",{"id":1108,"createTime":18,"updateTime":18,"relativeEntities":1110,"slug":18,"properties":1111,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1114,"statistic":18},[],{"title":1112},{"VI":1113},"Department of Mathematical Sciences, Indiana University South Bend, South Bend, USA",[],{"title":1116},{"VI":1117},"Anna K. Savvopoulou",{"id":1119,"sortIndex":82,"researcher":18,"roles":1120,"affiliations":1121,"properties":1130,"displayName":1132,"givenName":18,"familyName":18},"26839933-e596-42a5-8fe7-929769a841be",[116],[1122],{"id":1123,"sortIndex":19,"affiliation":1124,"properties":18},"879c7a73-b2f5-473b-bed7-d9af5185a373",{"id":1123,"createTime":18,"updateTime":18,"relativeEntities":1125,"slug":18,"properties":1126,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1129,"statistic":18},[],{"title":1127},{"VI":1128},"Department of Mathematics and Computer Science, Saint Mary’s College, Notre Dame, USA",[],{"title":1131},{"VI":1132},"Christopher M. Wedrychowicz",{"url":1101,"publisher":1134,"properties":1175},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":1135,"slug":10,"properties":1136,"entityType":16,"verifyStatus":17,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":19,"subjectFields":1139,"manageAffiliations":1144,"indexDatabases":1155,"url":18,"thumbnailPath":18,"statistic":1170,"gsStatistic":18,"type":85,"analyzePriority":18},[],{"issn":1137,"title":1138},{"VOID":13},{"VOID":15},[1140],{"id":22,"createTime":18,"updateTime":18,"relativeEntities":1141,"label":1142,"description":1143,"parentId":18,"standard":18,"scholarHubFieldId":18},[],{"EN":25},{},[1145,1150],{"id":29,"createTime":18,"updateTime":18,"relativeEntities":1146,"slug":18,"properties":1147,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1149,"statistic":18},[],{"title":1148},{"EN":33},[35],{"id":37,"createTime":18,"updateTime":18,"relativeEntities":1151,"slug":18,"properties":1152,"entityType":18,"verifyStatus":18,"verifyTime":18,"verifyNote":18,"languages":18,"translateLanguages":18,"viewCount":18,"url":18,"parentIds":1154,"statistic":18},[],{"title":1153},{"EN":41},[],[1156,1163],{"id":45,"indexDatabase":1157,"url":58,"indexYears":18,"academicFieldIds":1162,"indexDatabaseRanking":18},{"id":47,"createTime":18,"updateTime":18,"relativeEntities":1158,"label":1159,"description":1160,"key":54,"publicationTags":1161,"standard":18},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":1164,"url":73,"indexYears":74,"academicFieldIds":1169,"indexDatabaseRanking":77},{"id":64,"createTime":18,"updateTime":18,"relativeEntities":1165,"label":1166,"description":1167,"key":70,"publicationTags":1168,"standard":18},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76],{"impactFactor":19,"impactFactorByYear":1171,"i10Index":19,"i10IndexLast5Year":19,"totalPublication":80,"totalPublicationByYear":1172,"totalCitation":19,"totalCitationByYear":1173,"totalCitationPerPublication":19,"totalCitationPerPublicationByYear":1174,"hindexLast5Year":19,"hindex":19},{},{"2018":82,"2019":82},{},{},{"pages":1176,"volume":1178},{"VOID":1177},"51-64",{"VOID":1179},"37","2014-05-17",2014,[56,77]]