location:  Publications → journals → CMB
Abstract view

# On a Conjecture of Livingston

Published:2016-10-11
Printed: Mar 2017
• Siddhi Pathak,
Department of Mathematics and Statistics, Queen's University, Kingston, Canada, ON K7L 3N6.
 Format: LaTeX MathJax PDF

## Abstract

In an attempt to resolve a folklore conjecture of Erdös 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 $\bar{\mathbb{Q}}$ - 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 settle Erdös's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients will be needed to settle Erdös's conjecture for prime $q$.
 Keywords: non-vanishing of L-series, linear independence of logarithms of algebraic numbers
 MSC Classifications: 11J86 - Linear forms in logarithms; Baker's method 11J72 - Irrationality; linear independence over a field

 top of page | contact us | privacy | site map |