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On the regularized products of some Dirichlet series

Mounir Hajli

Abstract

In this paper, we show that the regularized determinants of some Dirichlet series are multiplicative. As an application, we give generalizations of Lerch's formula for the classical gamma function and we determine the sum of some Dirichlet series generalizing Euler's formula on the sum of the reciprocal of squares. We recover the results of Kurokawa and Wakayama, and give a new proof for some Euler's formulas.

On the regularized products of some Dirichlet series

Abstract

In this paper, we show that the regularized determinants of some Dirichlet series are multiplicative. As an application, we give generalizations of Lerch's formula for the classical gamma function and we determine the sum of some Dirichlet series generalizing Euler's formula on the sum of the reciprocal of squares. We recover the results of Kurokawa and Wakayama, and give a new proof for some Euler's formulas.
Paper Structure (5 sections, 10 theorems, 88 equations)

This paper contains 5 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. On the regularized determinant of some Dirichlet series
  3. On the variation of the regularized product $\mathop{\mathrm{\ooalign{$∏$\cr\hidewidth$∐$\hidewidth\cr}}}\limits_{k=0}^\infty \left( (k+x)^m+ y\right)$
  4. Some applications
  5. Appendix : On Lerch's formula

Key Result

Theorem 1.1

Let $m$ be a positive integer $\geq 2$. Let $L(x):=\mathop{\mathrm{\ooalign{$∏$\cr\hidewidth$∐$\hidewidth\cr}}}\limits_{k=0}^\infty(k+x)$, with $x\in \mathbb{C}$. We have where $\varepsilon\in \mathbb{R}$ such that $\varepsilon^2=1$, $\varepsilon^{\frac{1}{m}}$ is an $m$-th root of $\varepsilon$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • proof
  • proof : Proof of Theorem \ref{['gammam']}
  • Corollary 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 7 more