On the Lindel\"of hypothesis for general sequences

TL;DR

This paper investigates the Lindel"of hypothesis for general sequences, providing counterexamples to some conjectures, exploring the notion of 'generic' sequences, and extending results relating LH to the Riemann hypothesis in generalized number systems.

Contribution

It constructs counterexamples to the conjecture that LH holds for all admissible sequences and analyzes the probabilistic and topological nature of 'generic' sequences satisfying LH.

Findings

Counterexamples to the universal validity of LH for all admissible sequences.

Demonstration that LH can be true or false for 'generic' sequences depending on the interpretation.

Extension of the equivalence between the Riemann hypothesis and LH to Beurling generalized number systems.

Abstract

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindel\"of hypothesis (LH) for general sequences which coincides with the usual Lindel\"of hypothesis for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the ''generic'' admissible sequence of positive real numbers. In this paper, we give counterexamples to the first conjecture, and show that the second conjecture can be either true or false, depending on the meaning of ''generic'': we construct probabilistic processes producing sequences satisfying LH with probability 1, and we construct Baire topological spaces of sequences for which the subspace of sequences satisfying LH is meagre. We also extend the main result of Gonek, Graham, and Lee, stating that the Riemann hypothesis is equivalent to LH for the sequence of prime numbers, to the context of Beurling generalized number systems.