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Proceedings of the American Mathematical Society

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

 
 

 

The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis


Authors: Steven M. Gonek, Sidney W. Graham and Yoonbok Lee
Journal: Proc. Amer. Math. Soc. 148 (2020), 2863-2875
MSC (2010): Primary 11M06, 11M26
DOI: https://doi.org/10.1090/proc/14974
Published electronically: March 17, 2020
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Abstract: 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

$\displaystyle \sum _{ \substack {n\leq x \\ n\in \mathscr {N} }}n^{-it},$    

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
Email: gonek@math.rochester.edu

Sidney W. Graham
Affiliation: Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859
Email: sidney.w.graham@cmich.edu

Yoonbok Lee
Affiliation: Department of Mathematics, Research Institute of Basic Sciences, Incheon National University, 119 Academy-ro, Yeonsu-gu Incheon, 22012, Korea
Email: leeyb@inu.ac.kr, leeyb131@gmail.com

DOI: https://doi.org/10.1090/proc/14974
Keywords: Riemann hypothesis, Lindel\"of hypothesis
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
Article copyright: © Copyright 2020 American Mathematical Society