On the Number of Primes Less Than a Given Magnitude

November 1859, in the Berlin Academy's proceedings

Bernhard Riemann's paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" was first published in the Berlin Academy's Monatsberichte in November 1859.

He did not deliver it himself. He sent it to the Academy's secretary, Johann Encke, with a letter dated 19 October, and Ernst Kummer read it at the general meeting on 3 November.

Primes, moved into the complex plane

Riemann started from Euler's product identity relating primes to the zeta series, extended the zeta function into the complex domain, and derived a formula connecting prime-counting behaviour to its zeros.

He then used those zeros in an explicit formula whose terms describe fluctuations in the distribution of primes. He cited a comparison of Li(x) against prime counts that Gauss and Goldschmidt had carried out to x = 3,000,000.

The conjecture he put aside

Riemann conjectured that the relevant nontrivial zeros all lie on the line with real part one-half, and acknowledged that he lacked a rigorous proof.

He had made a few unsuccessful attempts and set the proof aside because it was unnecessary for his immediate objective.

The paper's subject was prime distribution; the conjecture is a brief, structurally important passage inside it, and the surviving manuscript is six pages long.