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I browsed Dirichlets Werke today and was kind of surprised by two remarks that he made on p. 354 (Über die Bestimmung ...) and p. 372 (Sur l'usage ...). In the second paper, he claims (my translation)

I have applied these principles to a demonstration of the remarkable formula given by Legendre for expressing in an approximate manner how many prime numbers there are below an arbitrary, but very large, limit.

In a handwritten note on the reprint he sent to Gauss he remarked that $\sum 1/\log n$ (this is Gauss's version of the PNT, at least if you replace the sum by an integral) is a better estimate than Legendre's.

I am a little bit puzzled as to why Dirichlet's claim to have proved the prime number theorem is not discussed anywhere in the literature. Or is it?

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Good question!! –  John Stillwell Mar 4 '10 at 22:37
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2 Answers

up vote 7 down vote accepted

Dirichlet's remark from the first paper is extracted and translated on page 98 of The Development of Prime Number Theory by Narkiewicz. So this has not passed completely unnoticed. Narkiewicz remarks that Dirichlet believed that his analytic methods would enable him to prove Legendre's conjecture, and that Dirichlet never returned to the problem.

Dirichlet remained interested in the asymptotic growth laws ("Asymptotische Gesetze") of arithmetic functions for the rest of his life, as seen from his 1849 paper with the estimate

$$ \sum_{n \leq x}d(n) = x\log(x) + (2\gamma - 1)x + O(x^{1/2}), $$

and a couple of other estimates, and a letter of 1858 to Kronecker reprinted in Dirichlet's Werke, where he mentions having obtained a substantial improvement of the error term $O(x^{1/2})$ by a new method.

Since Dirichlet demonstrably did not lose interest in such questions, and never returned to the PNT in print, it seems reasonable to believe that he discovered that his real-variable method would not yield the PNT.

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Narkiewicz, the "modern-day Dickson". I accepted this answer because of the reference - thanks! –  Franz Lemmermeyer Mar 9 '10 at 18:43
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The second paper is Sur l'usage des s\'eries infinies dan la th\'eorie des nombres Crelle $\mathbf{18}$ (1838), 259--274. The quote in Crelle is near the end, at the top of p. 272. After it he says he has determined some mean-value formulas for arithmetic functions (like sums of divisors) by similar techniques. The technique he is describing is that of encoding a sequence of interest as the coefficients in a Dirichlet series and then looking at its (real) pole. This method is indeed one of Dirichlet's important discoveries, but the prime number theorem is such an order of magnitude harder than the other results he lists here that he must have erroneously convinced himself that he could derive the prime number theorem by his new method just like he had derived other number-theoretic limit laws.

Amusingly, his notation for the Riemann zeta-function (on p. 272) is $\varphi(s)$!

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So, we have an apparent counterexample to the famous saying of Jacobi: Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is. Rather we learn it first from him. When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain ... Quoted in G Schubring, Zur Modernisierung des Studiums der Mathematik in Berlin, 1820-1840. –  John Stillwell Mar 5 '10 at 1:10
@John Stillwell: Certainly he never claimed that he proved the PNT, did he? –  Harry Gindi Mar 5 '10 at 6:21
Guess that's exactly what he was implying when he wrote "a demonstration of the remarkable formula given by Legendre". Nice catch, Professor Stillwell! –  J. H. S. May 6 '10 at 22:02
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