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.

  • 13
    $\begingroup$ The manuscript was submitted in 1943 to the American Mathematical Society’s Transactions, but withdrawn by Rademacher before publication.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) $\endgroup$ – Carlo Beenakker Feb 21 '18 at 16:18
  • 7
    $\begingroup$ 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 So I would imagine any copies of this withdrawn manuscript would have been well hidden, the AMS perhaps still has a copy in their files... $\endgroup$ – Carlo Beenakker Feb 21 '18 at 16:50
  • 1
    $\begingroup$ it's in the book linked to in the first comment; I copied the page, you can find it here $\endgroup$ – Carlo Beenakker Nov 15 '18 at 21:12
  • 1
    $\begingroup$ @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. $\endgroup$ – Todd Trimble Nov 17 '18 at 14:37
  • 1
    $\begingroup$ @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. $\endgroup$ – Todd Trimble Nov 17 '18 at 17:31

This does not directly answer the question, but gives the details of the story, including the source of the error.

Time Magazine, Monday April 30, 1945:
The year is erroneously given as 1943 in the book linked to below

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.

One day last month electrifying news arrived at the University of Chicago office of Dr. Adrian A. Albert, editor of the Transactions of the American Mathematical Society. A wire from the society's secretary, University of Pennsylvania Professor John R. Kline, asked Editor Albert to stop the presses: a paper disproving the Riemann hypothesis was on the way. Its author: Professor Hans Adolf Rademacher, a refugee German mathematician now at Penn.

On the heels of the telegram came a letter from Professor Rademacher himself, reporting that his calculations had been checked and confirmed by famed Mathematician Carl Siegel of Princeton's Institute for Advanced Study. Editor Albert got ready to publish the historic paper in the May issue. U.S. mathematicians, hearing the wildfire rumor, held their breath. Alas for drama, last week the issue went to press without the Rademacher article. At the last moment the professor wired meekly that it was all a mistake; on rechecking. Mathematician Siegel had discovered a flaw (undisclosed) in the Rademacher reasoning. U.S. mathematicians felt much like the morning after a phony armistice celebration. Sighed Editor Albert: ''The whole thing certainly raised a lot of false hopes."

The "undisclosed" flaw found by Siegel is identified on page 109 of The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics:


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.