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On the typical size and cancellations among the coefficients of some modular forms

Published online by Cambridge University Press:  19 April 2018

FLORIAN LUCA ,
MAKSYM RADZIWIŁŁ  and
IGOR E. SHPARLINSKI
FLORIAN LUCA
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa. e-mail: [email protected]
MAKSYM RADZIWIŁŁ
Affiliation:
Department of Mathematics, McGill University, Montreal, QC, H3A 0B9, Canada. e-mail: [email protected]
IGOR E. SHPARLINSKI
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia. e-mail: [email protected]
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Abstract

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽ n11/2(logn)−1/2+o(1) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.

In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 1 , January 2019 , pp. 173 - 189
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Banks, W. D. and Shparlinski, I. E. Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height. Israel J. Math. 173 (2009), 253–277.Google Scholar
[2] Barnet–Lamb, T., Geraghty, D., Harris, M. and Taylor, R.. A family of Calabi–Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47 (2011), 29–98.Google Scholar
[3] Canfield, E. R., Erdős, P. and Pomerance, C.. On a problem of Oppenheim concerning “Factorisatio Numerorum”. J. Number Theory 17 (1983), 1–28.Google Scholar
[4] Clozel, L., Harris, M. and Taylor, R. Automorphy for some l-adic lifts of automorphic mod l Galois representations. Pub. Math. IHES 108 (2008), 1–181.Google Scholar
[5] Deligne, P. La conjecture de Weil. Publ. Math. IHES 43 (1974), 273–307.Google Scholar
[6] Elliott, P. D. T. A.. Probabilistic number theory. II. Central limit theorems. Grundlehren Math. Wiss. 240 (Springer–Verlag, Berlin-New York, 1980).Google Scholar
[7] Elliott, P. D. T. A. Multiplicative functions and Ramanujan's -function. J. Austral. Math. Soc. Ser. A 30 (1980/81), 461–468.Google Scholar
[8] Elliott, P. D. T. A.. function, A central limit theorem for Ramanujan's tau. Ramanujan J. 29 (2012), 145–161.Google Scholar
[9] Elliott, P. D. T. A. and Kish, J.. Harmonic analysis on the positive rationals II: Multiplicative functions and Maass forms. Preprint, 2014 (available from http://arxiv.org/abs/1405.7132).Google Scholar
[10] Fouvry, É. and Michel, P.. Sommes de modules de sommes d'exponentielles. Pacific J. Math. 209 (2003), 261–288.Google Scholar
[11] Fouvry, É. and Michel, P.. Sur le changement de signe des sommes de Kloosterman. Ann. Math. 165 (2007), 675–715.Google Scholar
[12] Granville, A. and Soundararajan, K. Sieving and the Erdoős–Kac theorem. Equidistribution in number theory, an introduction. NATO Sci. Ser. II Math. Phys. Chem. 237 (2007), 15–27.Google Scholar
[13] Hall, R. R. and Tenenbaum, G. Divisors (Cambridge University Press, 1988).Google Scholar
[14] Harris, M., Shepherd-Barron, N. and Taylor, R.. A family of Calabi–Yau varieties and potential automorphy. Ann. Math. 171 (2010), 779–813.Google Scholar
[15] Hoffstein, J. and Ramakrishnan, D. Siegel zeros and cusp forms. Internat. Math. Res. Notices 6 (1995), 279–308.Google Scholar
[16] Iwaniec, H. Topics in classical automorphic forms. Amer. Math. Soc. (Providence, RI, 1997).Google Scholar
[17] Iwaniec, H. and Kowalski, E. Analytic number theory. Amer. Math. Soc. (Providence, RI, 2004).Google Scholar
[18] Koninck, J.-M. and Luca, F.. Analytic number theory: exploring the anatomy of integers. Amer. Math. Soc. (2012).Google Scholar
[19] Kowalski, E., Robert, O. and Wu, J. Small gaps in coefficients of L-functions and 𝔅-free numbers in short intervals. Rev. Mat. Iberoamericana 23 (2007), 281–326.Google Scholar
[20] Matomaki, K. and Radziwiłł, M. Sign changes of Hecke eigenvalues. Preprint, 2014 (available from http://arxiv.org/abs/1405.7671).Google Scholar
[21] Radziwiłł, M. and Soundararajan, K.. Moments and distribution of central L-values of quadratic twists of elliptic curves. Preprint, 2014 (available from http://arxiv.org/abs/1403.7067).Google Scholar
[22] Ramanujan, S. On certain arithmetical functions. Trans. Camb. Phil. Soc. 22 (1916), 159–184.Google Scholar
[23] Rouse, J. Atkin–Serre type conjectures for automorphic representations on GL(2). Math. Res. Lett. 14 (2007), 189–204.Google Scholar
[24] Sarnak, P. Some Applications of Modular Forms. (Cambridge University Press, 1990).Google Scholar
[25] Shparlinski, I. E. On the Sato–Tate conjecture on average for some families of elliptic curves. Forum Math. 25 (2013), 647–664.Google Scholar
[26] Shparlinski, I. E. On the Lang–Trotter and Sato–Tate conjectures on average for polynomial families of elliptic curves. Michigan Math. J. 62 (2013), 491–505.Google Scholar
[27] Suryanarayana, D. and Sitaramachandra Rao, R. The distribution of square-full integers. Arkiv för Matem. 11 (1973), 195–201.Google Scholar
[28] Tenenbaum, G. Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, 1995).Google Scholar
[29] Thorner, J. The error term in the Sato-Tate conjecture. Archiv der Math. 103 (2014), 147–156.Google Scholar