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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On an analogue of Titchmarsh’s divisor problem for holomorphic cusp forms
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by Nigel J. E. Pitt PDF
J. Amer. Math. Soc. 26 (2013), 735-776 Request permission

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
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3037786