Talk:Riemann zeta function

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Prime number density

All important results are based on the sieves. These are hard to do and there is only computational result. So to be honest the density of the prime numbers vanishes in the limit of all positive integers. But it does this only in the limit and for any finite and so big integer there is a finite prime number density in the positive integers! SteJaes (talk) 20:12, 16 March 2023 (UTC)[reply]

The discussion of the critical stripe and the critical line is incomplete!

The surface of the Riemann zeta function looks like this in the critical stripe:

All the zeros look like touches to the surface 0, like that there is a pencil pointing to 0, like dip, the curves look like roots or potency functions from the zero on the critical line to the borders of the stripe. I can offer some pictures showing that exemplary and fundamental behavior. On the borders of the critical stripe there are special behaviors. On the critical line the zero of the real and imaginary part coincide. For y=1 the real part does not have any zeros and the absolute function does not either. For y=0 the situation is different. It is like that the real and imaginary part do a schwebung and the absolute function is the upper limit of the schwebung without a zero possibly. This is hard to prove. The absolute function of the zeta function diverges at most.

This together gives reasons for the idea that the nontrivial zeros are isolated touches on the surface 0 in the Riemann zeta function.

SteJaes (talk) 20:37, 16 March 2023 (UTC)[reply]

Picture of the absolute function of the Riemann zeta function with the first two nontrivial zeros shown.

I suggest to improve the article with my picture of the first two nontrivial zeros. This shows exemplary and fundamental the behavior on the critical line in the critical stripe in the complex plane. This supports but does not prove the Riemann conjecture. It suggests on the other hand the path to a prove. SteJaes (talk) 20:48, 16 March 2023 (UTC)[reply]

Incorrect proof

I suppressed the (unsourced; and by the way the only appeal to Titchmarsh’s book is wrong) « Proof 2 » of the functional equation, which is incorrect for several reasons. In particular the series $\sum n^{s-1}$ in line 9 does in general not converge for $0<\Re (s)<1$ (neither to $\zeta (1-s)$ nor to anything else). Sapphorain (talk) 23:54, 21 December 2023 (UTC)[reply]

Thank you very much for spotting the error and removing the content. The alleged proof is a "modification of Titchmarsh's Fourier series proof" (see https://arxiv.org/abs/math/0305191).

If one proof is not enough, I guess one could add another proof – a candidate would be the (original) Fourier series proof from Titchmarsh, though I don't know if it makes sense to include it here – on the one hand, it should not be copied from Titchmarsh word for word; on the other hand, excluding the details that make the proof rigorous is not a good idea anyway. A1E6 (talk) 12:36, 5 April 2024 (UTC)[reply]

Finite sum of zeta(n)

There's nothing about the finite sum $\sum _{n=2}^{N}\zeta {(n)}=N-\sum _{k=1}^{\infty }{\frac {1}{k(k+1)^{N}}}$ . Wouldn't that be a useful addition? Where would it fit into the article? Renerpho (talk) 22:55, 18 June 2024 (UTC)[reply]

Before we can answer that question, we need to know: what is the context of the sum in published reliable sources that discuss it? —David Eppstein (talk) 23:06, 18 June 2024 (UTC)[reply]

I'll get back to that when I have a reference for you. An unrelated question: In Riemann zeta function#Infinite series, there's a formula

\sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\operatorname {Im} {\bigl (}(1+i)^{n}-(1+i)^{n}{\bigr )}={\frac {\pi }{4}}

. Isn't the left side equal to 0? Maybe it should be

(1+i)^{n}-(1-i)^{n}

? Renerpho (talk) 23:09, 18 June 2024 (UTC)[reply]

Okay, forget about that. I found it in the source, the formula as given in the article was wrong.[1] Renerpho (talk) 23:22, 18 June 2024 (UTC) No idea what this edit (5 June 2023) was supposed to achieve. Renerpho (talk) 23:32, 18 June 2024 (UTC)[reply]

"Duplication formula" for the gamma function should be "reflection formula"

I believe that in the proof of the functional equation, in the last step it says that one should apply the duplication formula for the Gamma function. I don't think that the duplication formula yields the result, though. The reflection formula together with the duplication formula, however, does seem to yield the result (or perhaps one. So, I believe that this is just a typo. As I'm not an expert, I wanted to check with others rather than changing it myself. NoahSD (talk) 23:51, 15 February 2025 (UTC)[reply]