Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms
Tóm tắt
In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and
$$r_{s, k}(n)$$
, the number of representations of a positive integer n as a sum of
$$s\;k$$
-th positive integral powers, based on the recently proved Main Conjecture in Vinogradov’s Mean Value Method.
Tài liệu tham khảo
Bourgain, J.: On the Vinogradov mean value. Tr. Mat. Inst. Steklova 296(2017), 36–46 (2016). arXiv:1601.08173
Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. 184(2), 633–682 (2016)
Bykovskiĭ, V.A.; Vinogradov, A.I.: Inhomogeneous convolutions(Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987), Anal. Teor. Chisel i Teor. Funktsiĭ, 8, 16–30, 296; translation in J. Soviet Math. 52 (1990), no. 3, 3004–3016
Deshouillers, J.-M., Iwaniec, H.: An additive divisor problem. J. Lond. Math. Soc. 26(1), 1–14 (1982)
Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982)
Drappeau, S.: Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method. Proc. Lond. Math. Soc. 114(4), 684–732 (2017)
Goldfeld, D.L.: Automorphic Forms and \(L\)-Functions for the Group \(GL(n, { R})\), with an Appendix by Kevin A. Broughan, vol. 99. Cambridge University Press, Cambridge (2006)
Hua, L.-K.: On Waring’s problem. Q. J. Math. Oxford 9, 199–202 (1938)
Ingham, A.E.: Some asymptotic formulae in the theory of numbers. J. Lond. Math. Soc. 2(3), 202–208 (1927)
Iwaniec, H.: Topics in Classical Automorphic Forms. Graduates Student Mathematics. AMS, New York (1997)
Iwaniec, H.: Spectral Methods of Automorphic Forms. Gradauate Student Mathematics. AMS, New York (2002)
Jiang, Y., Lü, G.: Shifted convolution sums for higher rank groups. Forum Math. 31(2), 361–383 (2019)
Linnik, YuV: The Dispersion Method in Binary Additive Problems. Translations of Mathematical Monographs. AMS, New York (1963)
Lü, G., Wu, J., Zhai, W.: Shifted convolution of cusp-forms with \(\theta \)-series. Ramanujan J. 40(1), 115–133 (2016)
Luo, W.: Shifted convolution of cusp-forms with \(\theta \)-series. Abh. Math. Semin. Univ. Hambg. 81(1), 45–53 (2011)
Miller, S.D.: Cancellation in additively twisted sums on \(GL(n)\). Am. J. Math. 128(3), 699–729 (2006)
Munshi, R.: Shifted convolution sums for \(GL(3) \times GL(2)\). Duke Math. J. 162(13), 2345–2362 (2013)
Ren, X., Ye, Y.: Hyper–Kloosterman sums of different moduli and their applications to automorphic forms for \(SL_{m}({ Z})\). Taiwan. J. Math. 20(6), 1251–1274 (2016)
Sun, Q.: Shifted convolution sums of \(GL_{3}\) cusp forms with \(\theta \)-series. Int. Math. Res. Not. IMRN 6, 1805–1829 (2017)
Vaughan, R.C.: The Hardy–Littlewood Method. Cambridge Tracts in Mathematics, vol. 125, 2nd edn. Cambridge University Press, Cambridge (1997)
Wooley, T.: The cubic case of the main conjecture in Vinogradov’s mean value theorem. Adv. Math. 294, 532–561 (2016)
Wooley, T.: The asymptotic formula in Waring’s problem. Int. Math. Res. Not. IMRN 7, 1485–1504 (2012)