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On a Conjecture of Livingston

Published online by Cambridge University Press:  20 November 2018

Siddhi Pathak
Siddhi Pathak*
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Kingston, ON K6L 3N6 e-mail: [email protected]
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Abstract

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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 $\left\{ -1,\,1 \right\}$, Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite $q\,\ge \,4$, highlighting that a new approach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime $q\,\ge \,3$, and indicate that more ingredients will be needed to settle Erdös conjecture for prime $q$.

Keywords

11J8611J72non-vanishing of L-serieslinear independence of logarithms of algebraic numbers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 184 - 195
Copyright
Copyright © Canadian Mathematical Society 2017

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