Riemann's: "On the Number of Prime Numbers less than a Given Quantity"
This is the first page of Bernhard Riemann's autograph of his article "On the Number of Prime Numbers less than a Given Quantity" ("Über die Anzahl der Primzahlen unter einer gegebenen Grösse."), which appeared in the Monatsberichte der Berliner Akademie, in November 1859. In that article Riemann proves that the zeta function, which became known as the "Riemann zeta function", has a meromorphic continuation to the whole complex plane, has a simple pole at s=1 and no other poles. Furthermore, Riemann relates the zeros of the zeta function to the number of primes that are less or equal to a given quantity, and states his famous conjecture: that all (non-trivial) zeros have real part 1/2. Riemann continues: "Certainly one would wish for a strict proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts ..."
Credits: The manuscript of Riemann's 1859 paper resides in the Manuscript Department of the Niederschsische Staats- und Universitetsbibliothek Gottingen (shelfmark: Cod. Ms. B. Riemann 3). The photos of Riemann's manuscript on this site are courtesy of the Gottingen Library.