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

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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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The question of whether the sum of Non-trivial Zeros of the Riemann Zeta Function is calculable is a highly debated topic in mathematics. At this time, there is no known closed form expression or algorithm for calculating this sum. However, there are some conjectures and theories that suggest it may be possible to calculate this sum.

One of the most famous conjectures related to the Riemann Zeta Function is the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann Zeta Function lie on the critical line Re(s) = 1/2. If this conjecture is true, then it may be possible to simplify the sum of Non-trivial Zeros of the Riemann Zeta Function in some way. However, the Riemann Hypothesis remains unproven and is considered one of the most challenging unsolved problems in mathematics.

In terms of convergence, it is known that the sum of Non-trivial Zeros of the Riemann Zeta Function diverges for certain values of x. This means that the sum does not converge to a finite value for all x. However, there are some cases where the sum may converge, such as when x is a positive integer. In general, the convergence of this sum is still an open question.

In conclusion, while there are some conjectures and theories that suggest the sum of Non-trivial Zeros of the Riemann Zeta Function may be calculable, there is currently no known method for calculating it. The topic remains an active area of research in mathematics, and further progress may be made in the future.
 

1. Can the sum of Non-trivial Zeros of the Riemann Zeta Function be calculated exactly?

No, the sum of Non-trivial Zeros of the Riemann Zeta Function cannot be calculated exactly. This is because the Riemann Zeta Function is an infinite series and the sum of Non-trivial Zeros cannot be expressed in a finite form.

2. Is there a method to approximate the sum of Non-trivial Zeros of the Riemann Zeta Function?

Yes, there are several methods to approximate the sum of Non-trivial Zeros of the Riemann Zeta Function. Some of these methods include the Riemann-Siegel formula, the Riemann-Siegel theta function, and the Riemann-Siegel series.

3. What is the significance of calculating the sum of Non-trivial Zeros of the Riemann Zeta Function?

The sum of Non-trivial Zeros of the Riemann Zeta Function is significant in studying the distribution of prime numbers. It also has applications in number theory, physics, and cryptography.

4. Are there any unresolved conjectures related to the sum of Non-trivial Zeros of the Riemann Zeta Function?

Yes, there are several unresolved conjectures related to the sum of Non-trivial Zeros of the Riemann Zeta Function. Some of these include the Riemann Hypothesis, the Lindelöf Hypothesis, and the Grand Riemann Hypothesis.

5. What progress has been made in calculating the sum of Non-trivial Zeros of the Riemann Zeta Function?

Over the years, mathematicians have made significant progress in approximating the sum of Non-trivial Zeros of the Riemann Zeta Function. Several methods and formulas have been developed, and new discoveries are continuously being made. However, the exact value of the sum remains an open problem in mathematics.

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