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Search: MSC category 11J72 ( Irrationality; linear independence over a field )

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1. CMB 2016 (vol 60 pp. 184)

Pathak, Siddhi
On a Conjecture of Livingston
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
Categories:11J86, 11J72

2. CMB 2008 (vol 51 pp. 32)

Choi, Stephen; Zhou, Ping
On Linear Independence of a Certain Multivariate Infinite Product
Let $q,m,M \ge 2$ be positive integers and $r_1,r_2,\dots ,r_m$ be positive rationals and consider the following $M$ multivariate infinite products \[ F_i = \prod_{j=0}^\infty ( 1+q^{-(Mj+i)}r_1+q^{-2(Mj+i)}r_2+\dots + q^{-m(Mj+i)}r_m) \] for $i=0,1,\dots ,M-1$. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space $\IQ F_0+\IQ F_1 +\dots + \IQ F_{M-1} + \IQ$ over $\IQ$ and show that among these $M$ infinite products, $F_0, F_1,\dots ,F_{M-1}$, at least $\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow \infty$.


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