Shifted euler constants and a generalization of euler-stieltjes constants

Abstract

The purpose of this article is twofold. First, we introduce the constants ζk(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 Lseries at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants γk(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 γ1(r/q) which was given by Blagouchine in 2015.