I remember reading somewhere that there is a (probably a family of) quick false proof of the Riemann hypothesis that starts by using complex logarithms in a bad way, then does some elementary calculations and out pops the result. A perusal of the General Mathematics section of the arXiv also shows an abundance of presumably false proofs of the same. These tend to be somewhat... unclear and I'd prefer avoiding to look at them in any detail.

Question: Can anyone describe a quick false proof of the Riemann hypothesis?

Preferably I'd like one that has a very clear false step that misuses complex logarithms or $n$-th roots or some such gadget in an identifiable way. This is both because of my curiosity, and because I figure I could use it as an exercise problem for undergrads (if I ever end up teaching a course on complex analysis) to explain why they should be careful about taking logarithms.

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    $\begingroup$ (It's easy to cheat: derive an arithmetic contradiction and use the principle of explosion. This is not enlightening in the desired way however.) $\endgroup$
    – anon
    Commented Aug 26, 2012 at 17:14
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    $\begingroup$ Couldn't there be an infinitely many number of ways NOT to prove RH? $\endgroup$ Commented Aug 26, 2012 at 19:42
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    $\begingroup$ There's a published paper with (essentially) the same title as this question, only in French: P. Cartier, Comment l'hypothèse de Riemann ne fut pas prouvée, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), pp. 35–48, Progr. Math., 22, Birkhäuser Boston, Boston, MA, 1982, MR0693308 (85f:11035). $\endgroup$ Commented Aug 26, 2012 at 23:56

1 Answer 1


Matthew Watkins has a collection of 'proofs' here.

Describe them would make you loose part of their flavor...
(errors are sometimes commented and the oldest and 'less serious' ;-) are at the end...)


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