## 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**

## 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