Non-vanishing of Dirichlet series with periodic coefficients

Abstract

For any periodic function f : N → C with period q, we study the Dirichlet series L(s, f ) := n≥1 f (n)/n s . It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1, where it has a simple pole with residueThus, the function is analytic at s = 1 when ρ = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L(1, f ) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L(1, f ).

A vanishing criterion for Dirichlet series with periodic coefficients

“…The previous two sections give a characterization of algebraic-valued periodic arithmetic functions, odd and even respectively whose L-series vanish at s = 1. In this section, we mention a theorem of Ram Murty and Tapas Chatterjee [4] which nicely ties the nonvanishing of the odd and the even functions to give us the characterization required. The proof in their paper is incorrect as care was not taken regarding the branch of logarithm.…”

“…Since c j (x) = c j (q − x), R belongs to the Z-module generated by (5). Indeed all relations R := q−1 x=1 C x a x = 0 in the Z-module generated by (4) satisfy C x = −C q−x , which along with the fact that C j (x) = C j (q − x) (which stem from the evenness of f ) imply that c j (x) = 0 ∀ 1 ≤ x ≤ q − 1. Thus, R is a Z-linear combination of (5).…”

Linear independence of harmonic numbers over the field of algebraic numbers

“…We state some simple applications of Baker's theorem as were done in [6], [9] and [11] to resolve the problem of transcendental nature of algebraic linear combination of π, logarithm of non-zero algebraic numbers and positive real units. We will use the next proposition due to Chatterjee and Gun to prove some of the theorems in the later section.…”

On a conjecture of Erdős and certain Dirichlet series

“…Their work was generalized by Gun, Murty and Rath [4] to the setting of algebraic number fields. The forthcoming paper [2] gives new proofs of some of the background results of this area. We also refer the reader to [12] for an expanded survey of the early history.…”