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Search: All articles in the CMB digital archive with keyword Logarithm

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1. CMB Online first

Pathak, Siddhi
 On a conjecture of Livingston In an attempt to resolve a folklore conjecture of ErdÃ¶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 $\bar{\mathbb{Q}}$ - 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 ErdÃ¶s'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 ErdÃ¶s's conjecture for prime $q$. Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbersCategories:11J86, 11J72

2. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
 Logarithms and the Topology of the Complement of a Hypersurface This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$. Keywords:Logarithm, homology groups and periodsCategories:32Q55, 14F45
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