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.
