In An older question I asked :
( See A Thue-Morse Zeta function (Generalized Riemann Zeta function and new GRH) )
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Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number, and recall that the Odiuos numbers are the indices of nonzero entries in the Thue-Morse sequence. Now consider the sequence of functions below:
$$f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+\dotsb$$
This is the zeta function valid for $\mathrm{Real}(s)>1$.
$$f(2,s)=1-2^{-s}+3^{-s}-4^{-s}+\dotsb$$
This is the alternating zeta function valid for $\mathrm{Real}(s)>0$.
$$f(3,s)=1-2^{-s}-3^{-s}+4^{-s}+5^{-s}-6^{-s}-7^{-s}+8^{-s}+\dotsb = 4^{-s} (\zeta(s,1/4) - \zeta(s,2/4) - \zeta(s,3/4) + \zeta(s,4/4) ) $$
( $\zeta(s,a)$ is Hurwitz zeta )
I'm not sure if this has an official name yet but it clear that it is valid for $\mathrm{Real}(s)>-1$. This sequence of functions is constructed in the similar way the Thue-Morse sequence is constructed.
$$\begin{align} &\vdots\\ f(\infty,s)&= \sum (-1)^{t_n} n^{-s} \end{align}$$
This is a nice generalization/variant of the Riemann Zeta function and the Dirichlet eta or Dirichlet $L$-functions. It follows that $f(m, s)$ is valid for $\mathrm{Real}(s)>-m+2$. Now there are two logical questions analogue to the questions about the Riemann Zeta function:
What are the functional equations for $f(m,s)$?
Call the $N^\text{th}$ zero $Z_n(m)$. Are all the zero's of $f(m,s)$ for any $m$ with $0<\mathrm{Real}(s)<1$ on the critical line $(\mathrm{Real}(Z_N(m))=1/2)$ ?
Is clearly a generalizations of the Riemann Hypothesis. And I think it might be true! (I made some plots that were convincing but the accuracy was low.)
I wonder if these functions have a name yet and what the answers to the 2 logical questions are. I also invite the readers to make more conjectures and variants with this.
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Some additional questions :
let $T(s) = f(\infty,s) $.
1) Is $T(s)$ meromorphic on The entire complex plane ?
2) how Many poles does $T(s)$ have ? Is it one ?
3) assuming 1) : What is The infinite product representation for $T(s)$ ? ( hadamard type product )
4) assuming 1),2) how fast is this function growing on The complex plane ? As fast as Riemann zeta ?? I assume so.
I think all of these are true. Maybe 2) can Be shown by induction from $f(n,z) $ To $f(n+1,z) $ ?? But infinity is no integer , so maybe not.