On a conjecture of Livingston

TL;DR

This paper disproves Livingston's conjecture for composite q ≥ 4, shows it holds for prime q ≥ 3, and emphasizes the need for new methods to resolve Erdős's conjecture on non-vanishing of certain L-series.

Contribution

The paper disproves Livingston's conjecture for composite q ≥ 4 and confirms it for prime q ≥ 3, revealing the complexity of Erdős's conjecture.

Findings

Disproved Livingston's conjecture for composite q ≥ 4.

Confirmed Livingston's conjecture for prime q ≥ 3.

Indicated the necessity of new approaches for Erdős's conjecture.

Abstract

In an attempt to resolve a folklore conjecture of Erd\H{o}s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the $\overline{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settles Erd\H{o}s's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients are needed to settle Erd\H{o}s's conjecture for prime $q$.