A vanishing criterion for Dirichlet series with periodic coefficients

Abstract

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.

Special values of derivatives of $L$-series and generalized Stieltjes constants

“…Therefore, by Conjecture 1, we get thatf k − ξf l = 0 on all natural numbers. This implies the Q-linear dependence of the functionsf k andf l and thus, contradicts the Q-linear independence of f k and f l by (4). Hence, the quotient L ′ (1, f k )/L ′ (1, f l ) is transcendental for all 1 ≤ k < l ≤ r, which in turn leads us to conclude that at most one of the numbers under consideration is algebraic.…”

“…The motivation for studying these constants emanates from our desire to understand special values of L-series. More precisely, when f is an arithmetical function, with period q, the Dirichlet series L(s, f ) := ∞ n=1 f (n) n s , has been the focus of intense study (see for example the survey article by Tijdeman [17] as well as [4], [8] and [15]). However, these papers studied the special value L(1, f ) whenever it is defined.…”

Linear independence of harmonic numbers over the field of algebraic numbers

“…many numbers which are linearly independent over Q. Finally rewriting (7) in terms of the linear combination of elements of the subset S and using Baker's theory, we can get at most ⌊ q−1 2 ⌋+1 2 many linear homogeneous equations in the variables c , i s with algebraic coefficients. So, altogether we get a linear homogeneous system of at most…”

“…many numbers which are linearly independent over Q. Finally rewriting (7) in terms of the linear combination of elements of the subset S and using Baker's theory, we can get at most…”

Transcendental numbers and special values of Dirichlet series

“…It may be fruitful to consider the slightly general framework suggested by Chowla. In this connection, Chatterjee, Murty and Pathak [9] have characterized all functions f for which L(1, f) = 0 in Chowla's problem. Related to this is a question (conjecture) of Erdös, that (13.1) does not vanish whenever f (n) = ±1 and f (q) = 0.…”