On the Lindelöf hypothesis for general sequences
Frederik Broucke, Sebastian Weishäupl
TL;DR
This work probes the Lindelöf hypothesis for general sequences $LH(\mathcal{N},M(x))$, revealing that a density-1 integer sequence can violate LH while probabilistic models yield LH almost surely for suitable $M(x)$ and that topological genericity can imply meagreness of the LH property. It shows that ‘generic’ behavior depends on the chosen notion of randomness or topology, and extends the LH–RH correspondence to Beurling generalized number systems, clarifying sharp error thresholds and constructing systems where LH behavior diverges from RH. Together, these results illuminate the sensitivity of LH to modeling choices and arithmetic structure, and widen the scope of LH to non-classical number systems.
Abstract
In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences which coincides with the usual Lindelöf 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.