Euler-Lehmer constants and a conjecture of Erdos

The Euler-Lehmer constants gamma(a, q) are defined as the limits lim(x ->infinity) ( Sigma(n <= x) 1/n - logx/q) n equivalent to a (mod q) We show that at most one number in the infinite list gamma(a,q), 1 <= a < q, q >= 2, is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdos. If f : Z/qZ -> Q is such that f(a) = +/- 1 and f(q) = 0, then Erdos conjectured that Sigma(infinity)(n=1) f(n)/n not equal 0. If q equivalent to 3 (mod 4), we show that the Erdos conjecture is true. (C) 2010 Elsevier Inc. All rights reserved.