Talk:Riemann zeta function

Clarification needed on approximate values of ordinates of zeros

The formula in the following referenced assertion, which has been recently inserted in the subsection « Other results » of «  Zeros, the critical line, and the Riemann hypothesis » is problematic, mainly because of the use by ‎178.219.5.13 of the symbol $\approx$, which is infamous for its mathematical imprecision. It is true that this symbol is used at different places in the paper (as well as $\simeq$), but on the one hand it is never defined there, and on the other hand the well-known symbol $\sim$, whose rigorous definition is very consensual amongst mathematicians, is also used, seemingly interchangeably with $\approx$ — and even with $\simeq$ !— (for instance the authors outrageously state the prime number theorem as $\pi (x)\approx Li(x)$, and mention the « asymptotic behaviour » $W(x)\approx \log x$ (!), although they constantly use the same symbol $\approx$ when rounding up numerical value to various decimals). Moreover the formula in the following referenced assertion is in fact not given under this form in the cited paper. What the authors state is that the ordinates $t_{n}$ of zeros on the critical line are given by a certain equation ((13) in the paper). Then, by ignoring the limit term in (13) (although pointing it is usually not zero), they consider a simplified equation (62), whose solutions (63)

${\widetilde {t}}_{n}=2\pi {\frac {n-{\frac {11}{8}}}{W({\frac {n-{\frac {11}{8}}}{e}})}}$

they state are approximate solutions for the ordinates of the Riemann zeros. They produce tables of computed values of $t_{n}$ and ${\widetilde {t}}_{n}$ showing these values look indeed close to each other, but they never rigorously say what they mean by « approximate ». All of this is extremely sloppy, and the shortcut adopted below makes things even sloppier.

(Reproduced insert, needing clarification/modification/(suppression?):

" The estimated imaginary part of the n-th zero on the critical line has this evaluation:

$\gamma _{n}\approx 2\pi {\frac {n-{\frac {11}{8}}}{W({\frac {n-{\frac {11}{8}}}{e}})}}\qquad$")

Sapphorain (talk) 14:06, 24 January 2023 (UTC)

If this description is accurate, it sounds too non-rigorous to include to me. —David Eppstein (talk) 18:40, 24 January 2023 (UTC)

I find the description accurate. I don't think there's disagreement that the formula is a good approximation, but I haven't found a paper with actual error bounds, asymptotic or otherwise. I asked about it on MathOverflow a few years back without any result. So it's interesting but probably not ready for inclusion. - CRGreathouse (t | c) 01:46, 25 January 2023 (UTC)

You got an answer there and it can only find the integer part 14.134725, 21.022040, 25.010858 ... 2A00:1370:8184:9B6:BC80:DC1C:77DA:2F65 (talk) 11:52, 26 January 2023 (UTC)

The answer there is still vague: "expected to have integer part correct": in what sense is "expected" used? Expected by whom? Is this proven? Conditionally proven? Conjectured? —David Eppstein (talk) 18:49, 26 January 2023 (UTC)

The nature of this formula is the same as similar formula that approximates Gram points. It is as the sentence describes asymptotically correct for all the zeros on the critical line. Its derivation in the origin is correct and the error term exists not for this estimation, but the original equation that this short equation is an estimation of. (That saying it is not impossible to derive the error term.) The reason it is here is that it is au pair with corresponding formula for Gram points and it is examined even beyond what Riemann zeta zeros are, and there is no similar short formula in the text. Words "approximate" and "asymptotical" used in the text are the exact description of the formula. If nothing the work deserves a link to it somewhere. Otherwise the imaginary parts look kind of magical and difficult to derive, while we do have at least some estimate. There is nowhere in the text any sort of general estimates for the values, while we have them. But, It is up to you. Sure, the authors came more from physics, but the formula is very sound and practical. 85.164.91.169 (talk) 18:15, 16 March 2023 (UTC)

References

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)

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)

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

riemann zeta surface

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)

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)

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)

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)

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)

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)

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)