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.

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 ).

The purpose of this article is twofold. First, we introduce the constants ζ k (α, r, q) where α ∈ (0, 1) and study them along the lines of work done on Euler constant in arithmetic progression γ(r, q) by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants γ k (χ) for a principal character χ. In particular we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant γ 1 (r/q) which was given by Blagouchine in 2015.

Abstract. In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function.

In 1965, A. Livingston conjectured the $\overline{\mathbb{Q}}$-linear independence of logarithms of values of the sine function at rational arguments. In 2016, S. Pathak disproved the conjecture. In this article, we give a new proof of Livingston’s conjecture using some fundamental trigonometric identities. Moreover, we show that a stronger version of her theorem is true. In fact, we modify this conjecture by introducing a co-primality condition, and in that case we provide the necessary and sufficient conditions for the conjecture to be true. Finally, we identify a maximal linearly independent subset of the numbers considered in Livingston’s conjecture.

ABSTRACT. Let f : Z/qZ → Z be such that f (a) = ±1 for 1 ≤ a < q, and f (q) = 0. Then Erdös conjectured that n≥1 f (n) n = 0. For q even, it is easy to show that the conjecture is true. The case q ≡ 3 ( mod 4) was solved by Murty and Saradha. In this paper, we show that this conjecture is true for 82% of the remaining integers q ≡ 1 ( mod 4).

Abstract. The goal of this article is twofold. We first extend a result of Murty and Saradha [7] related to the digamma function at rational arguments. Further, we extend another result of the same authors [8] about the nature of p-adic Euler-Lehmer constants. IntroductionFor a real number x = 0, −1, · · · , the digamma function ψ(x) is the logarithmic derivative of the gamma function defined bywhere γ is Euler's constant. Just like the case of the gamma function, the nature of the values of the digamma function at algebraic or even rational arguments is shrouded in mystery.In the rather difficult subject of irrationality or transcendence, sometimes it is more pragmatic to look at a family of special values as opposed to a single specific value and derive something meaningful. An apt instance here is the result of Rivoal [10] about irrationality of infinitude of odd zeta values as opposed to that of a single specific odd zeta value.In this context, Murty and Saradha [7] in a recent work have made some breakthroughs about transcendence of a certain family of digamma values. In particular, they proved the following. Theorem 1.1 (Murty and Saradha). For any positive integer n > 1, at most one of the φ(n) + 1 numbers in the setis algebraic.In this article, we extend their result and prove the following.2010 Mathematics Subject Classification: 11J91.

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