# A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence.

Let $m$ be a positive integer and $s$ a complex number.

Odiuos Number

Now consider the sequence of functions below

$f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$

This is the zeta function valid for $Real(s)>1$

$f(2,s)=1-2^{-s}+3^{-s}-4^{-s}+...$

This is the alternating zeta function valid for $Real(s)>0$

$f(3,s)=1-2^{-s}-3^{-s}+4^{-s}+5^{-s}-6^{-s}-7^{-s}+8^{-s}+...$

Im not sure if this has an official name yet but it clear that it is valid for $Real(s)>-1$.

This sequence of functions is constructed in the similar way the Thue-Morse sequence is constructed.

...

$f(\infty,s)= \sum (-1)^{t_n} n^{-s}$

This is imho a nice generalization/variant of the Riemann Zeta function and the Dirichlet eta or Dirichlet L-functions.

It follows that $f_m$ is valid for $Real(s)>-m+2$.

Now there are 2 logical questions analogue to the logical questions for the Riemann Zeta :

1) What are the functional equations for $f(m,s)$ ?

2) Are all the zero's of $f(m,s)$ (call the $N$'th zero $Z_n(m)$) for any $m$ with $0<Real(s)<1$ on the critical line $(Real(Z_N(m))=1/2)$ ?

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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