Is Selberg's Zeta Function Proof of the Riemann Hypothesis?

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SUMMARY

Selberg's zeta function does not provide a proof of the Riemann Hypothesis (RH). The relationship Z(s+1) = ζ(s)Z(s) suggests that if Z(s) has all roots with a real part of 1/2, RH would be true. However, the discussion concludes that RH remains unproven, as any valid proof would gain widespread attention. The connection between the Selberg trace and the Riemann-Weyl trace is noted, but it does not lead to a definitive proof of RH.

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  • Understanding of Selberg's zeta function
  • Familiarity with the Riemann zeta function (ζ(s))
  • Knowledge of geodesics and their relation to prime numbers
  • Basic concepts of the Riemann Hypothesis (RH)
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Did Selberg proved RH ??

the idea is that if the Selberg zeta function related to the length of Geodesics becomes

Z(s+1)= \zeta (s) Z(s)

and the Z(s) has all the roots with real part 1/2 would imply that RH is true ?? , sorry i am not an expert but found it curious since Riemann-Weyll trace is almost equal to Selberg trace and it is supposed that using his Selberg Trace we can prove that all the zeros lie on the line Re(s=1/2) , by the way the function multiplying Z(s) is the usual Riemann Zeta function. providing the surface of the laplacian has geodesic with length log(p_n) for every prime
 
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We would know if RH was proven, so the answer is 'No'. RH is so prominent, that it would spread like a wildfire.

I've seen a couple of ideas how to prove and even claims of proofs in the recent years. Some of them tried to apply physical ideas, a technique which I very much doubt will ever be successful.
 

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