Talk:Riemann zeta function

message: A formula for \zeta(s) involving the Thue-Morse sequence was recently added.  While this result is interesting, I have my doubts that it is sufficiently notable for an encyclopedia article on the Riemann zeta function.  I note that the same editor added similar content to the article on Apery's constant.  74.111.98.156 (talk) 16:57, 18 December 2022 (UTC)
thread_title:  Thue-Morse sequence 

message: 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)
replies: text: If this description is accurate, it sounds too non-rigorous to include to me. —David Eppstein (talk) 18:40, 24 January 2023 (UTC)
replies: text:  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)
replies: text: 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)
replies: text: 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)
replies: text:  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)
thread_title:  Clarification needed on approximate values of ordinates of zeros 

message: 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)
thread_title:  Prime number density 

message: 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)
thread_title:  The discussion of the critical stripe and the critical line is incomplete!