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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: florian.luca@wits.ac.za
MAKSYM RADZIWIŁŁ
Affiliation:
Department of Mathematics, McGill University, Montreal, QC, H3A 0B9, Canada. e-mail: maksym.radziwill@gmail.com
IGOR E. SHPARLINSKI
Affiliation:
Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia. e-mail: igor.shparlinski@unsw.edu.au
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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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