Some of the users here receive claimed proofs of the Riemann hypotheses on a regular bases. As fas as we know all of them have been wrong. But sometimes failure is also interesting.

So for all cases of proven wrong claimed proofs:

Have there been examples where the first obvious error appeared only at a late stage in the manuscript?

Are there examples which contain/imply some interesting*/remarkable/entertaining idea?

Maybe an idea which even was exploited otherwise?

Obviously the cases also can be categorised into cases from professionals and from laymen, and maybe in cases from both (e.g. from "outside" Mathematics, like Physics).

Are there interesting cases from both sides?

Are there repeating patterns which are simple to understand / communicate?

I think some answers which could be given also for this question, are found here ("Examples of interesting false proofs") on MO and indeed at least one answer contains a link to Peter Woit's blog named "not even wrong" where a few examples of wrong RH proofs are given and discussed.

This question focusses on RH in particular and since there should be many more claimed false proofs than these few examples it seems likely that there are also many more interesting cases than those mentioned.

Examples for related questions of a second class are:

Are you aware of interesting claimed proofs of the negation of RH?

Are any particularly interesting cases among them?

Are there interesting examples of claimed $\Re{\rho}\ne\frac{1}{2}$ cases?

Are there other claimed proofs of existence?

Again also here I am only interested in proven wrong proofs, so were someone has already pinpointed the mistake(s).


*)"interesting" is in this post supposed to mean interesting in terms of mathematics or mathematical considerations arising from.

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    $\begingroup$ Any "example of claimed $\Re(\rho) \ne 1/2$ cases" has to be material from a crank, of which the all-time winner for "interesting/remarkable/creative idea" is probably the author who submitted a paper claiming to show the Riemann zeta function is identically zero (no kidding, this really happened). As posed, this question needs more focus (if it is to be retained at all) since there is a ton of garbage on this theme which surely is not what you have in mind to hear about. Presumably you mean to focus on failed proofs or failed/incomplete strategies which generated interesting mathematics $\endgroup$
    – nfdc23
    Apr 6, 2018 at 12:16
  • 5
    $\begingroup$ to get started, here are some 60 proofs (and a few disproofs) of the Riemann hypothesis on vixra.org $\endgroup$ Apr 6, 2018 at 13:04
  • 1
    $\begingroup$ One of the better articles might be here: link.springer.com/chapter/10.1007%2F978-0-387-72126-2_8. As written there: George Pólya once had a young mathematician confide to him that he was working on the great Riemann Hypothesis. “I think about it every day when I wake up in the morning,” he said. Pólya sent him a reprint of a faulty proof that had been once been submitted by a mathematician who was convinced he’d solved it, together with a note: “If you want to climb the Matterhorn you might first wish to go to Zermatt where those who have tried are buried”. $\endgroup$
    – Todd Trimble
    Apr 6, 2018 at 17:38
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    $\begingroup$ What about creating a journal devoted to wrong proofs of RH? The referee's role would be to check that there's indeed an error (so as to qualify). One could even expect that it would then eventually accidentally to a correct (=mistakenly claimed wrong) proof of RH. $\endgroup$
    – YCor
    Apr 6, 2018 at 19:20
  • $\begingroup$ @YCor: My thoughts went in part in the same direction. For sure there wouldn't be a shortage of manuscripts. "Peer-review" though, might turn out being problematic. $\endgroup$ Apr 6, 2018 at 19:23


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