Dirichlet and the prime number theorem

Question

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?

Answer 1

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.

Answer 2

The second paper is Sur l'usage des séries infinies dans la théorie 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 $\varphi(s)$!