We address the question of non-vanishing of L(1, f ) where f is an algebraicvalued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions f for which L(1, f ) = 0. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.
In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function f , periodic with period q, taking values in {−1, 1} when q ∤ n and f (n) = 0 when q n, the series ∑ ∞ n=1 f (n) n does not vanish. This conjecture is still open in the case q ≡ 1 mod 4 or when 2φ(q) + 1 ≤ q. In this paper, we obtain the characteristic function of the limiting distribution of L(k, f ) for any positive integer k and Erdős function f with the same parity as k. Moreover, we show that the Erdős conjecture is true with "probability" one.2010 Mathematics Subject Classification. 11M99.
Abstract. In an attempt to resolve a folklore conjecture of Erdös regarding the nonvanishing at s = of the L-series attached to a periodic arithmetical function with period q and values in {− , }, Livingston conjectured theQ -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite q ≥ , highlighting that a new approach is required to settle ErdösâĂŹs conjecture. We also prove that the conjecture is true for prime q ≥ , and indicate that more ingredients will be needed to settle ErdösâĂŹs conjecture for prime q.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.