Abstract
The last twenty years of the nineteenth century witnessed a rapid progress in the theory of complex functions, summed up in the monumental treatises of Émile Picard1 (1891–1896) and Camille Jordan2(1893–1896). The development of the theory of integral functions, started by Karl Weierstrass (1876) and rounded up by Jacques Hadamard (1893), revived the interest in Riemann’s memoir and forced attempts to use these new developments to solve questions left open by Riemann.
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References
Picard, Charles Émile (1856–1941), Professor in Toulouse (1879–1881) and in Paris.
Jordan, Camille (1838–1922), Professor in Paris.
According to E.Landau (1903a) this name of the equivalent theorems 5.8 and 5.13 appears first in its German form Primzahlsatz in the thesis of H.von Schaper (1898).
According to S.Mandelbrojt (1967) Hadamard was actually in posession of Jensen’s formula before Jensen but did not not publish it, since he could not find for it any important application.
In fact its order equals 1.
Baillaud,Bourget 1905,II,452–457.
This paper of von Mangoldt is a good example of exactness and thoroughness in manipulating nonabsolutely convergent series and integrals. It largely exceeded the standard of his time, which can be observed by reading Cahen (1894) or Stieltjes (1885a, 1885b, 1889), and seems to be one of the first papers having these qualities.
According to Landau (1920) this result is due to Bohr alone.
von Mangoldt denoted this function by L(n). The notation ∧(n) and the name ‘Mangoldt’s function’ was introduced later, possibly first in Landau’s book (Landau 1909a), since previously Landau (e.g. in Landau 1903a) used the notation L(n) for it. This function has been used earlier for other purposes by Bougaieff (1888a,b) and Césaro (1888).
See the proof of Theorem 5.4.
“I proved for the first time ...that the function ζ(s) does not have roots of the form 1 + ßi. Mr Hadamard, before knowing about my research, also found the same theorem in a simpler way.”
Hadamard’s original proof omits this observation; the proof is so easy that possibly he regarded it as evident.
“We shall see that by modifying slightly the author’s analysis one can establish the same result rigorously.”
We have simplified here the notation used by Vallée-Poussin, who did not utilize the function ∧(n).
It seems that it was J.J.Sylvester (1892, footnote) who first observed that (5.30) is a consequence of the asymptotic equality of θ(x) and x.
Bernays, Isaak Paul (1888–1977), Professor in Göttingen and Zürich.
This theorem is an analogue for Dirichlet series of the Vivanti-Pringsheim theorem (Pringsheim 1894, Vivanti 1893), stating that any function whose Maclaurin series has positive coefficients must have a singularity at the intersection of the positive real half-line with its circle of convergence.
Fekete, Michael (1886–1957), Professor in Budapest and Jerusalem.
Wintner, Aurel Friedrich (1903–1964).
Here we simplify slightly, without changing the main idea, the original argument of de la Vallée-Poussin who performed his computation conserving all terms in the expressions considered.
Proofs may be found in Titchmarsh (1951), Chandrasekharan (1970).
Vinogradov, Ivan Matveevich (1891–1983), Professor in Moscow.
The proof of this result contains a small oversight, a correction of which can be found in a footnote on p. 272 of Landau (1912a).
Schmidt, Erhard (1876–1959), Professor in Berlin.
A simple proof is given in Landau (1910).
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Narkiewicz, W. (2000). The Prime Number Theorem. In: The Development of Prime Number Theory. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-13157-2_5
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