1. CMB Online first
 Pathak, Siddhi

On a conjecture of Livingston
In an attempt to resolve a folklore conjecture of ErdÃ¶s regarding
the nonvanishing 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:nonvanishing of Lseries, 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 $i=0,1,\dots ,M1$.
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
$\IQ F_0+\IQ F_1 +\dots + \IQ F_{M1} + \IQ$ over $\IQ$ and show that
among these $M$ infinite products, $F_0, F_1,\dots ,F_{M1}$, at least
$\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow
\infty$.
Category:11J72 
