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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.
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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 non-vanishing of L-series, linear independence of logarithms of algebraic numbers
MSC Classifications: 11J86, 11J72 show english descriptions Linear forms in logarithms; Baker's method
Irrationality; linear independence over a field
11J86 - Linear forms in logarithms; Baker's method
11J72 - Irrationality; linear independence over a field
 

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