Riemann publishes 'On the Number of Primes Less Than a Given Magnitude'

Background. Bernhard Riemann, a German mathematician, made profound contributions to analysis and number theory. His 1859 paper on the prime-counting function is regarded as a foundational paper of analytic number theory. It was the only paper he published on the subject of number theory, yet it contained ideas that influenced thousands of researchers. The paper studies the prime-counting function using analytic methods, consisting primarily of definitions, heuristic arguments, and sketches of proofs.

Publication Details. The paper, titled 'Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse', was published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin. It is a seminal 9-page paper. Riemann investigated the zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers. The work introduced the complex zeta function.

Mathematical Content. Riemann discussed the relationship between ζ(s) and the distribution of prime numbers, using the function J(x) as a measure for Stieltjes integration. He found a formula for the prime-counting function π(x), which he called F(x). His explicit formula for the number of primes states that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function.