# Where can I find Rademacher's wrong disproof of the Riemann Hypothesis?

Mathematical folklore has it that the famous algebraist Hans Rademacher once came up with a wrong disproof of the Riemann Hypothesis, which was initially believed by another famous mathematician, Carl Siegel. I vaguely remember that they say Rademacher's error was that he mistakenly assumed that logarithms of complex numbers are uni-valued.

But where can I find this particular work of Rademacher?

I would want to go over it in detail and I imagine something could be learned from it. A Google search didn't yield anything meaningful.

• The manuscript was submitted in 1943 to the American Mathematical Society’s Transactions, but withdrawn by Rademacher before publication. Here is the story (page 109) – Carlo Beenakker Feb 21 '18 at 16:18
• Another quote: Rademacher was terribly embarrased by the ordeal and never spoke of it again. [...] It was well known that no one was to mention the words "Riemann hypothesis" in his presence. So I would imagine any copies of this withdrawn manuscript would have been well hidden, the AMS perhaps still has a copy in their files... – Carlo Beenakker Feb 21 '18 at 16:50
• it's in the book linked to in the first comment; I copied the page, you can find it here – Carlo Beenakker Nov 15 '18 at 21:12
• @CarloBeenakker I am sorry to belabor this point, but I don't understand why you are calling it a disproof. Maybe you are using the word differently from how I'm using it. "... if you assume the Riemann Hypothesis is false [my emphasis], and then do this, this, and this, and you get a contradiction, then it must be true." So based on that, the conclusion would have been RH is true. In other words, based on the description that you quoted, it sounds as if Rademacher was trying to prove RH is true, via a proof by contradiction. Not a disproof by contradiction. – Todd Trimble Nov 17 '18 at 14:37
• @CarloBeenakker Thank you very much. I guess that should suffice for now, although I have to wonder if something got garbled in someone's transmission. – Todd Trimble Nov 17 '18 at 17:31

A sure way for any mathematician to achieve immortal fame would be to prove or disprove the Riemann hypothesis. This baffling theory, which deals with prime numbers, is usually stated in Riemann's symbolism as follows: "All the nontrivial zeros of the zeta function of s, a complex variable, lie on the line where $$\sigma$$ is ½ -- ($$\sigma$$ being the real part of s)." The theory was propounded in 1859 by Georg Friedrich Bernhard Riemann (who revolutionized geometry and laid the foundations for Einstein's theory of relativity). No layman has ever been able to understand it and no mathematician has ever proved it.