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

Abstract

For $s\in \mathbb{C}$ and $0<a<1$, let $\zeta (s,a)$ and ${\mathrm{Li}}_s(e^{2\pi ia})$ be the Hurwitz and periodic zeta functions, repectively. For $0<a\le 1/2$, put $Z(s,a):=\zeta (s,a)+\zeta (s,1-a)$, $P(s,a):={\mathrm{Li}}_s(e^{2\pi ia})+{\mathrm{Li}}_s(e^{2\pi i(1-a)})$, $Y(s,a):=\zeta (s,a)-\zeta (s,1-a)$ and $O(s,a):=-i({\mathrm{Li}}_s(e^{2\pi ia})-{\mathrm{Li}}_s(e^{2\pi i(1-a)}))$. 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.

16 pages. A remark in Section 5 is added