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There is a remark one can find in various books or survey articles (e.g., page 49 of Helmut Koch's "Number Theory: Algebraic Numbers and Algebraic Functions") saying Dirichlet figured out a proof of the unit theorem while listening to an Easter concert in the Sistine Chapel. My question is: what is the evidence for this story?

Today I did an internet search and found that Kummer wrote on p. 343 of volume 2 of Dirichlet's collected works that Dirichlet could work on math in all kinds of situations, and then Kummer says "Als Beispiel hierfür kann ich anführen, dass er die Lösung eines schwierigen Problems der Zahlentheorie, womit er sich längere Zeit vergeblich bemüht hatte, in der Sixtinischen Kapelle in Rom ergründet hat, während des Anhörens der Ostermusik, die in derselben aufgeführt zu werden pflegt" (translation: "As an example I can say that he found the solution to a difficult problem in number theory, which he had worked on for a considerable amount of time without success, in the Sistine Chapel in Rome while he was listening to the Easter music that tends to be played there.")

Notice Kummer does not say precisely what the "difficult problem" was. Maybe it is just an oral tradition that the problem is the unit theorem, but I would like a more definitive source.

I don't read German well, but if you do then Kummer's essay on Dirichlet can be read online. It starts on http://archive.org/stream/glejeunedirichl00dirigoog#page/n323/mode/1up and page 343 is http://archive.org/stream/glejeunedirichl00dirigoog#page/n355/mode/1up.

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    $\begingroup$ Helmut Koch's email address is available here. Koch is also the author of Dirichlet's entry on this page (giving the year 1844 as further information). Elstrodt's biographical sketch of Dirichlet also contains the episode and cites work of Dirichlet's student Bachmann [Ba.1] and [Ba.2] that might contain further pointers. $\endgroup$ – t.b. Aug 27 '12 at 3:57
  • $\begingroup$ OK, I wrote to Koch and Elstrodt. The paper [Ba.2] by Bachmann is in Crelle (vol. 67, pp. 200--204) and mentions nothing as it is purely a research article. $\endgroup$ – KCd Aug 27 '12 at 13:49
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    $\begingroup$ The question was posted on MO. $\endgroup$ – t.b. Sep 3 '12 at 11:52
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    $\begingroup$ You might want to repost this at History of Science and Mathematics Stack Exchange. $\endgroup$ – Joel Reyes Noche Sep 1 '16 at 7:11
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The following is a link to a PDF version of a paper published in the Clay Mathematics Proceedings by J. Elstrodt:

https://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf

I didn't see it referenced as a source here or in your posting on Mathematics Overflow, so I am providing it here.

Elstrodt's stated motivation for writing the paper is as follows: "The leading role of German mathematics in the second half of the nineteenth and even up to the faithful year 1933 would have been unthinkable without the foundations laid by Gauss, Jacobi, and Dirichlet. But whereas Gauss and Jacobi have been honoured by detailed biographies (e.g. \ldots), a similar account of Dirichlet's life and work is still a desideritum repeatedly ignored. \ldots The present account is a first attempt to remedy this situation.''

On page 26, the author states, ``According to C.G.J. Jacobi the unit theorem is one of the most important, but one of the thorniest of the science of number theory.' (references ommitted) adding,Kummer remarks that Dirichlet found the idea of proof when listening to Easter Music in the Sistene Chapel during his Italian journey.''' (and he cites the the 343 reference you alluded to by Kronecker)

Now, if one reads the previous two sentences of Elstrodt quickly, he is apt to conclude that the theorem of Dirichlet that the author alludes to is indeed the subject of the first sentence (the Unit Theorem); however, the second sentence does not quite confirm this. And, since the source Elstrodt provides as back-up, as you pointed out in your posting, not not provide convincing evidence as to exactly which theorem of Dirichlet is being referred to.

In any case, I think Elstrodt's 37-page paper may be worth reading as it provides other insights into Kummer's role as an authoritative source of information---this is because (and I am only briefly describing why), Ernst Kummer was married to a cousin of Dirichlet's wife, Rebecca Mendelssohn (sister of composer Felix Mendelssohn), which effectively made Kummer and Dirichlet cousins; whom by the way, had considerable interactions with Dirichlet until the latter's death. It is therefore quite likely that Kummer would have been able to provide many important details in the life of Dirichlet that other biographers would not who were not privy to such information.

Hopefully this in conjunction with the Oct., 2016 edit found on your posting at

https://mathoverflow.net/questions/106196/definitive-source-about-dirichlet-finally-proving-the-unit-theorem-in-the-sistin

adds some support, albeit not the definitive proof you were looking for, that the idea of a proof that Dirichlet had in mind while listening to music in the Sistene Chapel during his Italian journey may very well have pertained to the theorem we now know as Dirichlet's Unit Theorem.

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