The Euler-Lehmer constants γ (a, q) are defined as the limitsWe show that at most one number in the infinite listis an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdös. If f : Z/qZ → Q is such that f (a) = ±1 and f (q) = 0, then Erdös conjectured thatIf q ≡ 3 (mod 4), we show that the Erdös conjecture is true.