On the random Chowla conjecture
Tóm tắt
We show that for a Steinhaus random multiplicative function
$$f:{\mathbb {N}}\rightarrow {\mathbb {D}}$$
and any polynomial
$$P(x)\in {\mathbb {Z}}[x]$$
of
$$\deg P\ge 2$$
which is not of the form
$$w(x+c)^{d}$$
for some
$$w\in {\mathbb {Z}}$$
,
$$c\in {\mathbb {Q}}$$
, we have
$$\begin{aligned} \frac{1}{\sqrt{N}}\sum _{n\le N} f(P(n)) \xrightarrow {d} {{\mathcal {C}}}{{\mathcal {N}}}(0,1), \end{aligned}$$
where
$${{\mathcal {C}}}{{\mathcal {N}}}(0,1)$$
is the standard complex Gaussian distribution with mean 0 and variance 1. This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of
$$x\ge 1$$
, such that
$$\begin{aligned} \left| \sum _{n\le x} f(P(n))\right| \gg _{P} \sqrt{x} (\log \log x)^{1/2}, \end{aligned}$$
for any polynomial
$$P(x)\in {\mathbb {Z}}[x]$$
with
$$\deg P\ge 2,$$
which is not a product of linear factors (over
$${\mathbb {Q}}$$
). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear polynomial
$$P(n)=n,$$
where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be
$$O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon })$$
for any
$$\varepsilon >0$$
.
Tài liệu tham khảo
J. Basquin. Sommes friables de fonctions multiplicatives aléatoires. Acta Arith., (3)152 (2012), 243–266
E. Bombieri and J. Pila. The number of integral points on arcs and ovals. Duke Math. J., (2)59 (1989), 337–357
P. Borwein, S.K.K. Choi, and H. Ganguli. Sign changes of the Liouville function on quadratics. Canad. Math. Bull., 56(2) (2013), 251–257
J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat, and A. Sárközy. On finite pseudorandom binary sequences. IV. The Liouville function. II. Acta Arith., (4)95 (2000), 343–359
S. Chatterjee and K. Soundararajan. Random multiplicative functions in short intervals. Int. Math. Res. Not. IMRN, (3) (2012), 479–492
S. Chowla. The Riemann Hypothesis and Hilbert’s Tenth Problem. Mathematics and Its Applications, Vol. 4. Gordon and Breach Science Publishers, New York-London-Paris, (1965).
H. Davenport, D.J. Lewis, and A. Schinzel. Equations of the form \(f(x)=g(y)\). Quart. J. Math. Oxford Ser. (2), 12 (1961), 304–312
H. Davenport and A. Schinzel. Two problems concerning polynomials. J. Reine Angew. Math., (215)214 (1964), 386–391
P.D.T.A. Elliott. On the correlation of multiplicative functions. Notas Soc. Mat. Chile, (1)11 (1992), 1–11
P. Erdős. Some applications of probability methods to number theory. In: Mathematical Statistics and Applications, Vol. B (Bad Tatzmannsdorf, 1983), pp. 1–18. Reidel, Dordrecht, (1985).
M. Fried. On a conjecture of Schur. Michigan Math. J., 17 (1970), 41–55
A. Granville. \(ABC\) allows us to count squarefrees. Internat. Math. Res. Notices, (19) (1998), 991–1009
A. Granville. Smooth numbers: computational number theory and beyond. In: Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pp. 267–323. Cambridge University Press, Cambridge, (2008).
A. Granville and K. Soundararajan. Large character sums. J. Amer. Math. Soc., (2)14 (2001), 365–397
G. Halász. On random multiplicative functions. In: Hubert Delange colloquium (Orsay, 1982), Volume 83 of Publ. Math. Orsay, pp. 74–96. Univ. Paris XI, Orsay, (1983).
A.J. Harper. Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab., (2)23 (2013), 584–616
A.J. Harper. A note on the maximum of the Riemann zeta function, and log-correlated random variables. arXiv e-prints, page arXiv:1304.0677, (2013a).
A.J. Harper. On the limit distributions of some sums of a random multiplicative function. J. Reine Angew. Math., 678 (2013b), 95–124
A.J. Harper. Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos. Forum Math. Pi, 8 (2020), e1, 95
A.J. Harper. Almost Sure Large Fluctuations of Random Multiplicative Functions. International Mathematics Research Notices, 11 (2021), rnab299.
A.J. Harper and Y. Lamzouri. Orderings of weakly correlated random variables, and prime number races with many contestants. Probab. Theory Related Fields, (3-4)170 (2018), 961–1010
D.R. Heath-Brown. The density of rational points on curves and surfaces. Ann. of Math. (2), (2)155 (2002), 553–595
H.A. Helfgott and M. Radziwiłł. Expansion, divisibility and parity. arXiv e-prints, page arXiv:2103.06853, (2021).
C. Hooley. On binary quartic forms. J. Reine Angew. Math., 366 (1986), 32–52
C. Hooley. On another sieve method and the numbers that are a sum of two \(h\)th powers. II. J. Reine Angew. Math., 475 (1996), 55–75
B. Hough. Summation of a random multiplicative function on numbers having few prime factors. Math. Proc. Cambridge Philos. Soc., (2)150 (2011), 193–214
Y.-K. Lau, G. Tenenbaum, and J. Wu. On mean values of random multiplicative functions. Proc. Amer. Math. Soc., (2)141 (2013), 409–420
D. Mastrostefano. An almost sure upper bound for random multiplicative functions on integers with a large prime factor. Electron. J. Probab., 27 (2022), Paper No. 32, 1–21
K. Matomäki, M. Radziwiłł, and T. Tao. An averaged form of Chowla’s conjecture. Algebra Number Theory, (9)9 (2015), 2167–2196
J. Maynard and Z. Rudnick. A lower bound on the least common multiple of polynomial sequences. Riv. Math. Univ. Parma (N.S.), (1)12 (2021), 143–150
D.L. McLeish. Dependent central limit theorems and invariance principles. Ann. Probability, 2 (1974), 620–628
J. Najnudel. On consecutive values of random completely multiplicative functions. Electron. J. Probab., 25 (2020), Paper No. 59, 28
N. Ng. The distribution of the summatory function of the Möbius function. Proc. London Math. Soc. (3), (2)89 (2004) 361–389
V.V. Prasolov. Polynomials, volume 11 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, (2004). Translated from the 2001 Russian second edition by Dimitry Leites.
G. Reinert and A. Röllin. Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab., (6)37 (2009), 2150–2173
A. Schinzel. Reducibility of polynomials in several variables. II. Pacific J. Math., (2)118 (1985), 531–563
K. Soundararajan and M.W. Xu. Central limit theorems for random multiplicative functions. arXiv e-prints, page arXiv:2212.06098, (2022).
T. Tao. The logarithmically averaged Chowla and Elliott conjectures for two-point correlations. Forum Math. Pi, 4 (2016) e8, 36
T. Tao and J. Teräväinen. Odd order cases of the logarithmically averaged Chowla conjecture. J. Théor. Nombres Bordeaux, (3)30 (2018), 997–1015
T. Tao and V. Vu. Additive combinatorics, Volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (2006).
T. Tao. The logarithmically averaged Chowla and Elliott conjectures for two-point correlations. In: Forum of Mathematics, Pi, Volume 4. University Press, (2016).
J. Teräväinen. On the Liouville function at polynomial arguments. arXiv e-prints, page arXiv:2010.07924, (2020).
V.Y. Wang and M.W. Xu. Paucity phenomena for polynomial products (2022). arXiv:2211.02908.
A. Wintner. Random factorizations and Riemann’s hypothesis. Duke Math. J., 11 (1944), 267–275