Talk:The Riemann zeta function and tuning

Review needed

This page is linked from almost every important edo, and I would consider it "high priority". Despite the length of the page and numerous derivations, some questions and problems remain:

• The construction itself remains largely unmotivated. Why the specific error functions? There is a large amount of handwaving and heuristics. We can use different cyclic error functions which will not lead to zeta, but they seem equally valid. If there is no specific reason to use cosine functions with these specific weights, then this should be clearly mentioned. Currently it seems like the page is actively trying to obscure these facts to make it seem like the connection is more "natural" than it actually is.

• Why focus specifically on the critical strip? The page currently states

As s approaches the value s = 1/2 of the critical line, the information content, so to speak, of the zeta function concerning higher primes increases [...]

This is very vague and should be clearly explained.

• The derivation starts from the 'naive' definition of the zeta function, which does not converge for s < 1. At some point it switches to the analytic continuation. It is not clear why or how this is valid for this application (although empirically it seems to work out). I should note that being careless about such things is what leads people to ridiculous claims such as 1+2+3+... = -1/12.

Finally, I believe there are still a lot of inaccuracies, although those may be easily fixed. I am not an expert on this topic, it would be great to have the page checked by someone who actually is an expert on complex analysis or the like.

– Sintel🎏 (talk) 18:01, 5 April 2025 (UTC)