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

• zetafunction
In summary, the conversation discusses the possibility of proving the Riemann Hypothesis (RH) using the Selberg zeta function and the Selberg trace. It is suggested that if the Selberg zeta function has all its roots with a real part of 1/2, then RH would be true. However, the speaker also mentions that there have been attempts to prove RH in recent years, but none have been successful so far.

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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

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.

## 1. What is the Riemann Hypothesis (RH)?

The Riemann Hypothesis (RH) is one of the most famous unsolved problems in mathematics. It was proposed by Bernhard Riemann in 1859 and states that all non-trivial zeros of the Riemann zeta function lie on the critical line of 0.5+it, where t is a real number.

## 2. Who is Selberg and what is his contribution to RH?

Atle Selberg was a Norwegian mathematician who made significant contributions to number theory and analysis. In 1942, he proved a partial result towards RH, known as the Selberg's zeta function. However, he did not prove the full Riemann Hypothesis.

## 3. Did Selberg prove RH?

No, Selberg did not prove the Riemann Hypothesis. However, his work on the Selberg's zeta function was an important step towards understanding the Riemann zeta function and its connection to prime numbers.

## 4. What is the current status of RH?

The Riemann Hypothesis is still an unsolved problem in mathematics. Many mathematicians have made attempts to prove or disprove it, but so far, no one has been able to provide a complete proof.

## 5. Why is the Riemann Hypothesis important?

The Riemann Hypothesis has important implications in number theory, especially in the distribution of prime numbers. If it is proven to be true, it would have far-reaching consequences in many areas of mathematics, including cryptography and physics. It is also considered one of the most challenging and famous unsolved problems in mathematics.