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


Author: Nigel J. E. Pitt
Journal: J. Amer. Math. Soc. 26 (2013), 735-776
MSC (2010): Primary 11F11, 11F30; Secondary 11F72, 11N37
DOI: https://doi.org/10.1090/S0894-0347-2012-00750-4
Published electronically: October 11, 2012
MathSciNet review: 3037786
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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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Additional Information

Nigel J. E. Pitt
Affiliation: Departamento de Matemática, Universidade de Brasília, DF 70910-900, Brazil
Email: pitt@mat.unb.br

DOI: https://doi.org/10.1090/S0894-0347-2012-00750-4
Keywords: Titchmarsh divisor problem, Möbius randomness, Kloosterman sums, Kuznetsov formula, shifted convolution
Received by editor(s): August 8, 2011
Received by editor(s) in revised form: November 18, 2011, and April 30, 2012
Published electronically: October 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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