Finding new ways to express the Rieman zeta function

Question

I would like to ask your opinion about the use that it has to find new ways to express the ζ function.Could it help us to prove or disprove the Riemann hypothesis?I have a proof thay i will be posting soon for a new way to express ζ but first i want your opinion thanks

Answer 1

There is a mathematical question : what can we do and can we prove the RH with all those different formulas for $\zeta(s)$ ?

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• Of course a proof of the Riemann hypothesis will be based on a new ways to think about $\zeta(s)$ and the primes

• Your new representation of $\zeta(s)$ should not exist for those functions very similar to $\zeta(s)$ but for which the RH fails. So it will have to be based on the Euler product and the functional equation.

  In particular, you should check if your new representation and argument in favor of the RH works the same way for $$F(s) =\alpha L(s,\chi_5)+\overline{\alpha} L(s,\overline{\chi_5})$$ (where $\chi_5$ is the complex character modulo $5$) having the same functional equation as $\zeta(s)$ but no Euler product. Automatically it means there are no chances to prove the RH this way, because $F(s)$ has many zeros off the critical line.