# Shifted Euler constants and a generalization of Euler-Stieltjes constants

## TL;DR

This paper introduces new shifted Euler constants and generalizes Euler-Stieltjes constants, providing new formulas and applications in number theory, including evaluations of integrals and Dirichlet L-series at critical points.

## Contribution

It defines and studies shifted Euler constants $\zeta_k(\alpha,r,q)$ and generalizes Euler-Stieltjes constants, extending their applications and providing new closed-form expressions.

## Key findings

- Defined shifted Euler constants and studied their properties.
- Connected constants to integrals involving Dirichlet divisor problem error terms.
- Provided a new proof for a closed-form of the first generalized Stieltjes constant.

## Abstract

The purpose of this article is twofold. First, we introduce the constants $\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $\gamma(r,q)$ 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 $\gamma_k(\chi)$ for a principal character $\chi.$ 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 $\gamma_1(r/q)$ which was given by Blagouchine in 2015.