The values of zeta functions composed by the Hurwitz and periodic zeta functions at integers

TL;DR

This paper investigates the special values of functions derived from Hurwitz and periodic zeta functions at integers, revealing their rationality, polynomial structure, and connections to spectral densities of Gaussian distributions.

Contribution

It provides explicit descriptions of the values of these functions at integers, including rationality, polynomial forms, and their relation to exponential functions, extending understanding of their algebraic and analytic properties.

Findings

Values at negative integers are rational numbers.

Values at positive integers are polynomials of trigonometric functions with rational coefficients.

The functions relate to spectral densities of certain Gaussian processes.

Abstract

For $s \in {\mathbb{C}}$ and $0 < a <1$, let $\zeta (s,a)$ and ${\rm{Li}}_s (e^{2\pi ia})$ be the Hurwitz and periodic zeta functions, repectively. For $0 < a \le 1/2$, put $Z(s,a) := \zeta (s,a) + \zeta (s,1-a)$, $P(s,a) := {\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s (e^{2\pi i(1-a)})$, $Y(s,a) := \zeta (s,a) - \zeta (s,1-a)$ and $O(s,a):= -i \bigl( {\rm{Li}}_s (e^{2\pi ia}) - {\rm{Li}}_s (e^{2\pi i(1-a)}) \bigr)$. Let $n \ge 0$ be an integer and $b := r/q$, where $q>r>0$ are coprime integers. In this paper, we prove that the values $Z(-n,b)$, $\pi^{-2n-2} P(2n+2,b)$, $Y(-n,b)$ and $\pi^{-2n-1} O(2n+1,b)$ are rational numbers, in addition, $\pi^{-2n-2} Z(2n+2,b)$, $P(-n,b)$, $\pi^{-2n-1} Y(2n+1,b)$ and $O(-n,b)$ are polynomials of $\cos (2\pi/q)$ and $\sin (2\pi/q)$ with rational coefficients. Furthermore, we show that $Z(-n,a)$, $\pi^{-2n-2} P(2n+2,a)$, $Y(-n,a)$ and $\pi^{-2n-1} O(2n+1,a)$ are polynomials of $0<a<1$ with rational coefficient, in addition, $\pi^{-2n-2} Z(2n+2,a)$, $P(-n,a)$, $\pi^{-2n-1} Y(2n+1,a)$ and $O(-n,a)$ are rational functions of $\exp (2 \pi ia)$ with rational coefficients. Note that the rational numbers, polynomials and rational functions mentioned above are given explicitly. Moreover, we show that $P(s,a) \equiv 0$ for all $ 0 < a < 1/2$ if and only if $s$ is a negative even integer. We also prove similar assertions for $Z(s,a)$, $Y(s,a)$, $O(s,a)$ and so on. In addition, we prove that the function $Z(s,|a|)$ appears as the spectral density of some stationary self-similar Gaussian distributions.