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Integral representations of the Riemann zeta function of odd argument

Jean-Christophe Pain

Abstract

In this article we obtain, using an expression of the digamma function $ψ(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $ζ(2p+1)$ for any positive value of $p$. The integrand consists of the product of a polynomial by one or two elementary trigonometric functions. Examples for the first values of the argument are given. Some of them were already derived by other methods.

Integral representations of the Riemann zeta function of odd argument

Abstract

In this article we obtain, using an expression of the digamma function due to Mikolas, integral representations of the zeta function of odd arguments for any positive value of . The integrand consists of the product of a polynomial by one or two elementary trigonometric functions. Examples for the first values of the argument are given. Some of them were already derived by other methods.

Paper Structure

This paper contains 3 sections, 3 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Examples
  3. Conclusion

Key Result

Theorem 1.1

We have, for $p\in\mathbb{N}^*$: with where

Theorems & Definitions (6)

  • Theorem 1.1
  • proof
  • Lemma 1.2
  • proof
  • Corollary 1.2.1
  • proof