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

Florian Luca, Maksym Radziwiłł, Igor E. Shparlinski

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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)| ≤ n 11/2 (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.

Original languageEnglish (US)
Pages (from-to)173-189
Number of pages17
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number1
StatePublished - Jan 1 2019

ASJC Scopus subject areas

  • General Mathematics


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