Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis
HTML articles powered by AMS MathViewer

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
  • J. Brian Conrey and Amit Ghosh, Remarks on the generalized Lindelöf hypothesis, Funct. Approx. Comment. Math. 36 (2006), 71–78. MR 2296639, DOI 10.7169/facm/1229616442
  • J. Pintz, Oscillatory properties of $M(x)=\sum _{n\leq x}\mu (n)$. I, Acta Arith. 42 (1982/83), no. 1, 49–55. MR 678996, DOI 10.4064/aa-42-1-49-55
  • E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
  • Paul Turán, On a new method of analysis and its applications, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. With the assistance of G. Halász and J. Pintz; With a foreword by Vera T. Sós; A Wiley-Interscience Publication. MR 749389
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11M06, 11M26
  • Retrieve articles in all journals with MSC (2010): 11M06, 11M26
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