The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis

Abstract

We recast the classical Lindelöf hypothesis as an estimate for the sums ∑ n ≤ x n − i t \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 ∑ n ≤ x n ∈ N n − i t , \begin{equation*} \sum _{ \substack {n\leq x \\ n\in \mathscr {N} }}n^{-it}, \end{equation*} where N \mathscr {N} can be a quite general sequence of real numbers. We support this with several examples and show that when N = P \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 N = P ( a , q ) \mathscr {N}=\mathbb {P}(a, q) , the primes congruent to a ( mod q ) a \pmod q , with a a coprime to q q , then the Riemann hypothesis holds for all Dirichlet L L -functions with characters modulo q 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.