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
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 \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.References
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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: gonek@math.rochester.edu
- Sidney W. Graham
- Affiliation: Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859
- MR Author ID: 76030
- 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
- MR Author ID: 857291
- Email: leeyb@inu.ac.kr, leeyb131@gmail.com
- 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: https://doi.org/10.1090/proc/14974
- MathSciNet review: 4099775