Is the sum of Non-trivial Zeros of the Riemann Zeta Function calculable?

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SUMMARY

The discussion centers on the calculability of the sum of Non-trivial Zeros of the Riemann Zeta Function, specifically the function f(x) defined as f(x) = ∑ exp(ρ x), where the sum is taken over all Non-trivial zeros (ρ) of ζ(s). It is established that this sum converges to f(x) for every x, except at points of discontinuity. Additionally, under the assumption of the Riemann Hypothesis (RH), the result may simplify, and it is confirmed that the series ∑ log(ζ(ns)) for s > 1 converges to a finite value.

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  • Understanding of Non-trivial Zeros of the Riemann Zeta Function
  • Familiarity with complex analysis and convergence of series
  • Knowledge of the Riemann Hypothesis (RH)
  • Basic concepts of logarithmic functions in relation to zeta functions
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  • Research the implications of the Riemann Hypothesis on zeta functions
  • Study convergence criteria for series involving complex functions
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Mathematicians, number theorists, and researchers interested in analytic number theory and the properties of the Riemann Zeta Function.

tpm
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Can this sum be made??

let be the sum:

[tex]f(x) = \sum_{\rho}exp(\rho x)[/tex]

where the sum is made over all Non-trivial zeros of [tex]\zeta (s)[/tex]

is the sum 'calculable' i mean:

* the sum converges to the function f(x) for every x (even x big) except perhaps at certain points where f(x) has discontinuities

* If we asume RH then does the result simplifies ??... thanks.

Also i would like to know if [tex]\sum_{n=1}^{\infty} log ( \zeta (ns)[/tex] s >1 converges to a finite value.
 
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2nd question, Yes.
 

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