Zeros and the functional equation of the quadrilateral zeta function

In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z(s,a):=\zeta (s,a) + \zeta (s,1-a)$ with $1/4 \le a \le 1/2$ are on only the non-positive even integers exactly same as in the case of $(2^s-1) \zeta (s)$. We also prove that all real zeros of the bilateral periodic zeta function $P(s,a):={\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s (e^{2\pi i(1-a)})$ with $1/4 \le a \le 1/2$ are on only the negative even integers just like $\zeta (s)$. Moreover, we show that all real zeros of the quadrilateral zeta function $Q(s,a):=Z(s,a) + P(s,a)$ with $1/4 \le a \le 1/2$ are on only the negative even integers. On the other hand, we prove that $Z(s,a)$, $P(s,a)$ and $Q(s,a)$ have at least one real zero in $(0,1)$ when $0<a<1/2$ is sufficiently small. The complex zeros of these zeta functions are also discussed when $1/4 \le a \le 1/2$ is rational or transcendental. As a corollary, we show that $Q(s,a)$ with rational $1/4 < a < 1/3$ or $1/3 < a < 1/2$ does not satisfy the analogue of the Riemann hypothesis even though $Q(s,a)$ satisfies the functional equation that appeared in Hamburger's or Hecke's theorem and all real zeros of $Q(s,a)$ are located at only the negative even integers again as in the case of $\zeta (s)$.Comment: 12 pages. We changed the title. Some typos are correcte

Additional file 4 of Transcriptomic population markers for human population discrimination

: Infinium Human OmniExpressExome microarray. (DOCX 11 kb

Additional file 10 of Transcriptomic population markers for human population discrimination

: Table S4. List of mRNA transcripts and TLDA probes. (DOCX 14 kb

Additional file 7 of Transcriptomic population markers for human population discrimination

: RNA isolation procedure. (DOCX 10 kb

Additional file 3 of Transcriptomic population markers for human population discrimination

: Figure S2. A binary Decision-Tree classifier built based on UTS2 and UGT2B17 data (left Panel) and for UTS2 (Right Panel) obtained from Caucasian (n = 37), and Chinese (n = 29) blood samples. (DOCX 87 kb

Additional file 2 of Transcriptomic population markers for human population discrimination

: Figure S1. The location of optimal hyperplane (black line) and supporting vectors (yellow lines) determined based on SVM method. (DOCX 42 kb

Additional file 8 of Transcriptomic population markers for human population discrimination

: Microarray analysis. A detailed description of Microarray statistical analysis. (DOCX 33 kb