Riemann's 1859 Manuscript
Riemann's paper, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the number of primes less than a given quantity), by Bernhard Riemann, was first published in the Monatsberichte der Berliner Akademie, in November 1859. This paper, just six manuscript pages in length, introduced radically new ideas to the study of prime numbers — ideas which led, in 1896, to independent proofs by Hadamard and de la Vallée Poussin of the prime number theorem. This theorem, first conjectured by Gauss when he was a young man, states that the number of primes less than x is asymptotic to x/log(x). Very roughly speaking, this means that the probability that a randomly chosen number of magnitude x is a prime is 1/log(x).
Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function, defined by ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ... . He also formulated a conjecture about the location of these zeros, which fall into two classes: the "obvious zeros" -2, -4, -6, etc., and those whose whose real part lies between 0 and 1. Riemann's conjecture was that the real part of the nonobvious zeros is exactly 1/2. That is, they all lie on a specific vertical line in the complex plane.
Riemann checked the first few zeros of the zeta function by hand. They satisfy his hypothesis. By now over 1.5 billion zeros have been checked by computer. Very strong experimental evidence. But in mathematics we require a proof. A proof gives certainty, but, just as important, it gives understanding: it helps us understand why a result is true.
Why is the Riemann hypothesis interesting? The closer the real part of the zeros lies to 1/2, the more regular the distribution of the primes. To draw a statistical analogy, if the prime number theorem tells us something about the average distribution of the primes along the number line, then the Riemann hypothesis tells us something about the deviation from the average.
© Clay Mathematics Institute 2005 except for Riemann's 1859 manuscript, used by permission of Niedersächsische Staats- und Universitätsbibliothek Göttingen and its transcription and translation, used by permission of theDavid Wilkins.