Contribution of Halphen to the PNT

Question

In A COLLECTION OF MODERN MATHEMATICAL CLASSICS ANALYSIS (1961) edited by Richard Bellman, the editor writes the following in the introduction to the second paper:

Paper 2: "A new solution of Warring's problem" by G. H. Hardy and J. E. Littlewood (1920)

"[...] but it remained for Hadamard and De La Vallee Poussin, aided by a remarkabke representation due to Halphen, to establish the Prime Number Theorem."

Question: what is the "representation due to Halphen" Bellman is talking about?

Thanks.

Answer 1

A discussion of G.H. Halphen's contribution to the attack on the Prime Number Theorem is given by Wladyslaw Narkiewicz's book, "The Development of Prime Number Theory: From Euclid to Hardy and Littlewood", pg.157ff:

Riemann's extension of the zeta-function to complex arguments was unsuccessfully used by G.H. Halphen (1883) to prove the asymptotic relation $$ \vartheta(x) = \sum_{p\le x} \log p = (1+o(1))x, \tag{4.22} $$ a result which can be easily shown to be equivalent to the Prime Number Theorem.

Halphen utilised the expression of the sum $\sum_{n \le x} a_n$ of coefficients of an arbitrary Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$ by the integral $$ \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} f(s) \frac{x^s}{s} \mathrm{d}s, $$ which in the special case $$ a_n = \begin{cases} 1/k & \text{if $n=p^k$ with prime $p$} \\ 0 & \text{if $n$ is not a prime power} \end{cases}$$ appeared already in Riemann's memoir. (See Kronecker 1878.) A correct proof of this formula under rather weak assumptions was later given by O. Perron (1908).

References

Halphen, G.H. (1883): Sur l'approximation des sommes de fonctions numériques. Comptes Rendus Acad. Sci. Paris, 96, 634-637. [Oevres, IV, 96-98, Gauthier-Villars, Paris 1924.

Halphen, G.H. (1885): Notice sure les travaux mathématiques. Gauthier-Villars, Paris. [Oevers, I, 1-47, Gauthier-Villars, Paris 1916.]

Narkiewicz, Wladyslaw (2000): The Development of Prime Number Theory: From Euclid to Hardy and Littlewood, Springer Science & Business Media, 449 pages

Perron, O. (1908): Zur Theorie der Dirichetschen Reihne, J. Reine Angew. Math., 134, 95-143.