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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis
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by Steven M. Gonek, Sidney W. Graham and Yoonbok Lee PDF
Proc. Amer. Math. Soc. 148 (2020), 2863-2875 Request permission


We recast the classical Lindelöf hypothesis as an estimate for the sums $\sum _{n\leq x}n^{-it}$. This leads us to propose that a more general form of the Lindelöf hypothesis may be true, one involving estimates for sums of the type \begin{equation*} \sum _{ \substack {n\leq x \\ n\in \mathscr {N} }}n^{-it}, \end{equation*} where $\mathscr {N}$ can be a quite general sequence of real numbers. We support this with several examples and show that when $\mathscr {N}=\mathbb {P}$, the sequence of prime numbers, the truth of our conjecture is equivalent to the Riemann hypothesis. Moreover, if our conjecture holds for $\mathscr {N}=\mathbb {P}(a, q)$, the primes congruent to $a \pmod q$, with $a$ coprime to $q$, then the Riemann hypothesis holds for all Dirichlet $L$-functions with characters modulo $q$, and conversely. These results suggest that a general form of the Lindelöf hypothesis may be both true and more fundamental than the classical Lindelöf hypothesis and the Riemann hypothesis.
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Additional Information
  • Steven M. Gonek
  • Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
  • MR Author ID: 198665
  • Email:
  • Sidney W. Graham
  • Affiliation: Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859
  • MR Author ID: 76030
  • Email:
  • Yoonbok Lee
  • Affiliation: Department of Mathematics, Research Institute of Basic Sciences, Incheon National University, 119 Academy-ro, Yeonsu-gu Incheon, 22012, Korea
  • MR Author ID: 857291
  • Email:,
  • Received by editor(s): November 26, 2018
  • Received by editor(s) in revised form: September 16, 2019, and December 4, 2019
  • Published electronically: March 17, 2020
  • Additional Notes: The third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIP) (No. 2016R1C1B1008405).
    The third author is the corresponding author
  • Communicated by: Amanda Folsom
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 2863-2875
  • MSC (2010): Primary 11M06, 11M26
  • DOI:
  • MathSciNet review: 4099775