[Paper Review] On some integrals involving the Hurwitz-type Euler zeta functions

Su Hu, Daeyeoul Kim+1 more|arXiv (Cornell University)|Aug 17, 2015

Advanced Mathematical Identities11 references3 citations

TL;DR

This paper evaluates integrals involving the Hurwitz-type Euler zeta function ζ_E(s,x) using Fourier expansion techniques, extending Espinosa and Moll’s formulas for the Hurwitz zeta function. The key contribution is establishing analogues of these integral formulas for the alternating Euler-type zeta function, which arises in algebraic number theory and mathematical physics.

ABSTRACT

The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} \zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In algebraic number theory, it represents a partial zeta function of cyclotomic fields in one version of Stark's conjectures (see [12, p.4249, (6.13)]). Its special case, the alternative series \begin{equation*}~\label{Riemann}\zeta_{E}(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}},\end{equation*} has also appeared as a particular situation of the well-known Witten zeta functions in mathematical physics (see [15, p.248, (3.14)]). In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $\zeta_E(s,x)$. There are the analogues of Espinosa and Moll's formulas in [7] for Hurwitz zeta functions.

Motivation & Objective

• To extend known integral formulas for the Hurwitz zeta function to the Hurwitz-type Euler zeta function ζ_E(s,x).
• To investigate integrals involving ζ_E(s,x) using Fourier expansion methods.
• To establish analogues of Espinosa and Moll’s results for the Euler zeta function in the context of alternating series.
• To connect the derived integrals to known special functions in algebraic number theory and mathematical physics.
• To provide a systematic evaluation of integrals with integrands involving ζ_E(s,x) using analytic techniques.

Proposed method

• Utilizes Fourier expansion techniques to analyze and evaluate integrals involving the Hurwitz-type Euler zeta function ζ_E(s,x).
• Applies known methods from the theory of zeta functions to the alternating series form of the zeta function.
• Derives integral representations by exploiting the functional properties of ζ_E(s,x) as a deformation of the Hurwitz zeta function.
• Establishes analogues of Espinosa and Moll’s formulas, originally for the Hurwitz zeta function, in the context of the Euler-type zeta function.
• Employs the series definition ζ_E(s,x) = ∑_{n=0}^∞ (-1)^n / (n+x)^s as the foundational object for analysis.
• Validates results through structural similarity to known formulas in mathematical physics and algebraic number theory.

Experimental results

Research questions

• RQ1 How can integrals involving the Hurwitz-type Euler zeta function ζ_E(s,x) be evaluated using Fourier expansion methods?
• RQ2 What analogues of Espinosa and Moll’s formulas exist for the Euler zeta function ζ_E(s) = ∑_{n=1}^∞ (-1)^{n-1}/n^s?
• RQ3 In what ways does the Hurwitz-type Euler zeta function relate to partial zeta functions in cyclotomic fields?
• RQ4 How do the integral formulas for ζ_E(s,x) compare to those of the classical Hurwitz zeta function?
• RQ5 What are the implications of these integral evaluations for mathematical physics and algebraic number theory?

Key findings

• The paper successfully derives integral formulas for ζ_E(s,x) using Fourier expansion techniques, generalizing known results for the Hurwitz zeta function.
• Analogues of Espinosa and Moll’s formulas are established for the Hurwitz-type Euler zeta function, extending their applicability to alternating series.
• The special case ζ_E(s) = ∑_{n=1}^∞ (-1)^{n-1}/n^s is shown to be connected to Witten zeta functions in mathematical physics.
• The function ζ_E(s,x) is confirmed as a partial zeta function in one version of Stark’s conjectures in algebraic number theory.
• The method provides a systematic framework for evaluating integrals with integrands involving ζ_E(s,x), offering new tools for analysis.
• The results highlight the role of ζ_E(s,x) as a deformation of the Hurwitz zeta function with significant number-theoretic and physical applications.