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On Dirichlet Series Involving $ζ(s)$ and Extensions of the Euler-Mascheroni Constant

Takumi Noda

TL;DR

This work develops a unified analytic framework for Dirichlet series built from the Riemann zeta-function by introducing $D(s_1,s_2;\lambda)$ and a generalized Euler constant $\gamma(\alpha,\beta;\mu)$. It proves an integral representation for $D$, derives numerous closed-form evaluations at integer arguments in terms of $\zeta$-values and the Bendersky constants $A_m$, including explicit expressions for $D(1,1;\mu)$ and related constants. A Lerch-transcendent–based functional relation (and accompanying index-shifting identities) connects these series to classical special functions, enriching the structural understanding of $\zeta$-involving Dirichlet series. Together, the results offer a systematic toolkit for explicit evaluation and structural study of such series and their arithmetic constants, with historical links to Goldbach and Euler's constants.

Abstract

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of the Euler--Mascheroni constant and express certain special values explicitly in terms of the Bendersky constants. These results provide a unified framework for evaluating Dirichlet series involving the Riemann zeta-function at integer arguments, together with the associated number-theoretic constants.

On Dirichlet Series Involving $ζ(s)$ and Extensions of the Euler-Mascheroni Constant

TL;DR

This work develops a unified analytic framework for Dirichlet series built from the Riemann zeta-function by introducing and a generalized Euler constant . It proves an integral representation for , derives numerous closed-form evaluations at integer arguments in terms of -values and the Bendersky constants , including explicit expressions for and related constants. A Lerch-transcendent–based functional relation (and accompanying index-shifting identities) connects these series to classical special functions, enriching the structural understanding of -involving Dirichlet series. Together, the results offer a systematic toolkit for explicit evaluation and structural study of such series and their arithmetic constants, with historical links to Goldbach and Euler's constants.

Abstract

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of the Euler--Mascheroni constant and express certain special values explicitly in terms of the Bendersky constants. These results provide a unified framework for evaluating Dirichlet series involving the Riemann zeta-function at integer arguments, together with the associated number-theoretic constants.
Paper Structure (8 sections, 12 theorems, 78 equations)

This paper contains 8 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation.
  4. Further Proposition and Examples
  5. Proofs of Theorems 2.1 and 2.2
  6. Proof of Theorem 2.3
  7. Proofs of Theorems 2.4 -- 2.6
  8. Functional relations

Key Result

Theorem 2.1

Let $\lambda \in \mathbb{C}$ with $\lambda \notin \mathbb{Z}_{< 0}$. The series ${D}(s_1, s_2; \lambda)$ in $(1.1)$ converges absolutely for any $s_1,s_2 \in \mathbb{C}$ except when $s_1\in \{0, -1, -2, \dots\}$. Furthermore, the following integral representation holds: This representation also defines a holomorphic function on the entire $(s_1, s_2)$-plane except when $s_1 \in \{0, -1, -2, \dots

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3: A relation between the generalized Euler constant and the Bendersky constants
  • Theorem 2.4: A relation for the Stieltjes constants
  • Theorem 2.5
  • Proposition
  • Theorem 2.6
  • Theorem 2.7: A functional relation for $D(s_1,s_2;\lambda)$
  • Proposition 3.1
  • proof
  • ...and 16 more