Mathematics · 1859
Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse
Royal Prussian Academy of Sciences, Berlin · November 1859
Extends the zeta function to complex values, relates its zeros to the count of primes and states the Riemann hypothesis.
Read the text The whole poster as a web page
- Print from
- A2
- License
- Public domain
- Source
- de.wikisource.org
- Retrieved
- 27 September 2026
- Language
- German
- Rights holder
- Bernhard Riemann, 1859
Theme
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Edition
The paper as printed in the Monatsberichte der Königlichen Preuss. Akademie der Wissenschaften zu Berlin, aus dem Jahre 1859 (Berlin, 1860), pp. 671-680. The text comes from the proofread transcription of German Wikisource (raw wikitext of the pages Seite:RiemannPrim1859.djvu/1 to /10), checked line by line and formula by formula against two scans of that printing: Commons, File:RiemannPrim1859.djvu (from Google Books j-IaAAAAYAAJ), and the Internet Archive item monatsberichted02unkngoog. The Monatsberichte print no title: the title is the one of Riemann’s Gesammelte mathematische Werke (1876, p. 136), and the sentence of the monthly report that introduces the paper is in the header, with the heading of its session; the heading of the report and the item before it (Steiner) are left out. Differences from Wikisource, all back to the print: the long s of the ligature “ſs” is normalized to a round s, giving “ss” where Wikisource writes “ß” (Grösse, dass, lässt, Gauss); “ausdrücken” (p. 673, Wikisource “audrücken”); no colon after “in die beiden folgenden” (p. 676); the two misprints of the Monatsberichte that Wikisource corrects are kept as printed, “(x)ψ” for ψ(x) in the first integral of p. 673 and the missing sum sign before the second-last logarithm on p. 676; the notes of Wikisource are left out. Letterspaced names are letterspaced. The formulas are set with KaTeX, Gauss’s function Π as a letter as printed, and the line “p^-s durch ..., p^-2s durch ...” of p. 675, a display line in the print that Wikisource sets as text, is set as a display. The closing quotation mark of the title in the report is “”” as printed.
Rights
Bernhard Riemann died on 20 July 1866 (Wikidata Q42299) and the paper was published in 1860, in the Monatsberichte for 1859: it is in the public domain in France, where the author died before 1 January 1956, and in the United States, where it was published before 1 January 1931. Wikimedia Commons marks the scan as public domain (PD-old-100-expired).
