Given a number field, the Euler–Kronecker constant is defined as the constant term in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function at the point s= 1 . In the case of real and imaginary quadratic fields, a closed-form expression for the Euler–Kronecker constants can be obtained with the help of suitable Kronecker limit formulas. In this article, we avoid the use of Kronecker limit formulas and derive an explicit series representation of these constants with the help of an asymptotic series representation of the coefficients appearing in the Laurent series expansion of the Dedekind zeta function at s= 1 . As a result, the expressions obtained do not require evaluation of the special functions appearing in the Kronecker limit formulas.

In this paper, we prove a criterion for the irrationality of certain constants which arise from the Ramanujan summation of a family of infinite divergent sums. As an application, we provide a sufficient criterion for the irrationality of the values of the Riemann zeta function in the interval (0, 1). We further see that our discussion leads to a natural generalization of a result of Sondow on the irrationality criterion for the Euler-Mascheroni constant.

In this article, we show that the Riemann hypothesis for an Lfunction F belonging to the Selberg class implies that all the derivatives of F can have at most finitely many zeros on the left of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974

We derive a series representation of the generalized Stieltjes constants which arise in the Laurent series expansion of partial zeta function at the point s =1. In the process, we introduce a generalized gamma function and deduce its properties such as functional equation,Weierstrass product and reflection formulas along the lines of the study of a generalized gamma function introduced by Dilcher in 1994. These properties are used to obtain a series representation for the k-th derivative of Dirichlet series with periodic coefficients at the point s =1. Another application involves evaluation of a class of infinite products of which a special case is an identity of Ramanujan.

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 L-series 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.

A famous identity of Gauss gives a closed form expression for the values of the digamma function Ψ(x) at rational arguments x in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdo[double-acute]s which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for nth Catalan number in terms of these functions.

In this article we consider a generalization of Stieltjes constants and study its relation with special values of certain Dirichlet series. Further we show a connection of these constants with a generalization of Digamma function. Some of the results obtained are a natural generalization of the identities of Gauss, Lehmer, Dilcher and many other authors.