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$\def\Re{\operatorname{Re}}$ We start with

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\qquad \Re(s)>1\tag{1}$$

$$\zeta(1-s)=\sum_{n=1}^{\infty}\frac{1}{n^{1-s}}\qquad \Re(s)<0\tag{2}$$

$$\eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n^s}\qquad \Re(s)>0\tag{3}$$

$$\eta(1-s)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{1-s}}\qquad \Re(s)<1\tag{4}$$

We will define the symmetrized sums for $\zeta$ and $\eta$ functions as: $$f(s)=\sum_{n=1}^{\infty}\left(\frac{1}{n^s}+\frac{1}{n^{1-s}}\right)\tag{5}$$

$$g(s)=\sum_{n=1}^{\infty}\left(\frac{(-1)^n}{n^s}+\frac{(-1)^n}{n^{1-s}}\right)\tag{6}$$

Their partial sums are: $$f_m(s)=\sum_{n=1}^{m}\left(\frac{1}{n^s}+\frac{1}{n^{1-s}}\right)\tag{7}$$

$$g_m(s)=\sum_{n=1}^{m}\left(\frac{(-1)^n}{n^s}+\frac{(-1)^n}{n^{1-s}}\right)\tag{8}$$

It is known that in the critical strip $0<\Re(s)<1$, the following 4 things are identical: (A) $\zeta(s)=0$; (B) $\zeta(1-s)=0$; (C) $\eta(s)=0$; (D) $\eta(1-s)=0$.

Denote $Z(F)$ the set of zeros for function $F(z)$.

Question (1) Are $f(s)$ and $g(s)$ convergent in the critical strip $0<\Re(s)<1$? (it is obvious that $g(s)$ is convergent)

Question (2) Can we prove that, in the critical strip $0<\Re(s)<1$, $Z(\zeta)\leq Z(f) $, $Z(\eta)\leq Z(g) $.

Question (3) If (1) and (2) are proved, is it easier to study the zero distribution for $f_m(s)$ and $g_m(s)$ than to study the zero distribution for $f(s)$ and $g(s)$?

Any comments and references are welcomed! -mike

EDIT: Here are some plots of $f(3,s)$ and $g(3,s)$. enter image description here enter image description here

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  • $\begingroup$ Both terms in the series for $f(s)$ diverge for $0\leq s \leq 1$. It is not obvious that $g$ converges. Only when $s$ is real is this obvious. $\endgroup$ – Winther Nov 28 '14 at 2:26

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