On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms

Pitt, Nigel

doi:10.1090/S0894-0347-2012-00750-4

On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms
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by Nigel J. E. Pitt

J. Amer. Math. Soc. 26 (2013), 735-776

DOI: https://doi.org/10.1090/S0894-0347-2012-00750-4

Published electronically: October 11, 2012

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Abstract:

The Fourier coefficients $a(n)$ of a holomorphic cusp form for the modular group are considered at values $n=p-1$ for primes $p$ up to $X$, and their sum shown to be smaller than the trivial bound by a power of $X$. The same bound is also shown to hold for the sum of $\mu (n)a(n-1)$ for natural numbers $n$ up to $X$, where $\mu$ denotes the Möbius function. The proofs require establishing non-trivial bounds for sums of Kloosterman sums and shifted convolutions of the coefficients which are better in the ranges required than known estimates. These are then used to bound bilinear forms in $a(mn-1)$, which in conjunction with previous work of the author, slightly corrected here, proves the main results.

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