The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis

Gonek, Steven; Graham, Sidney; Lee, Yoonbok

doi:10.1090/proc/14974

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.

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