Bernhard Riemann presents in August 1859, on the occasion of his election as a corresponding member of the Berlin Academy of Sciences, a paper of barely eight pages titled "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Primes Less Than a Given Magnitude"). In it he extends the zeta function — studied by Euler only for real values — to the complex plane, and formulates, as an incidental one-sentence remark within the paper, the conjecture that would bear his name: that all non-trivial zeros of the zeta function have real part equal to 1/2. The Riemann hypothesis, closely linked to the distribution of prime numbers, remains unproven more than 165 years later, being one of the seven Millennium Prize Problems of the Clay Mathematics Institute (together with P versus NP, the only one of the seven still open). Atle Selberg and Paul Erdős themselves would develop in 1949 an elementary proof of the prime number theorem without relying on the hypothesis, while the Bombieri-Vinogradov theorem (1965) offers, in many applications, a partial substitute for the still-unproven generalized Riemann hypothesis.