Riemann Zeta Function
A complex function central to analytic number theory; the location of its zeros encodes the distribution of the prime numbers.
The Riemann zeta function is one of the most studied objects in mathematics. For complex numbers s with real part greater than 1 it is defined by the infinite series:
ζ(s) = 1 + 1/2s + 1/3s + 1/4^s + …
For these values the series converges absolutely. The function can be extended analytically to the entire complex plane except for a single pole at s = 1, and it is this extended function that is meant when mathematicians speak of "the zeta function."
The connection to the primes is given by the Euler product, discovered by Leonhard Euler in the 18th century:
ζ(s) = Π over primes p of 1/(1 − p^−s)
The identity holds because every integer factors uniquely into primes, and it establishes the first deep link between a smooth analytic object and the irregular sequence of prime numbers. Euler used it to prove that the sum of the reciprocals of the primes diverges, a stronger statement than the infinitude of primes.
Riemann's 1859 memoir showed that the function satisfies a functional equation relating ζ(s) to ζ(1 − s), which reveals its behavior across the whole plane. The zeta function has zeros at the negative even integers −2, −4, −6, … (the "trivial" zeros). All other zeros — the "nontrivial" zeros — lie in the critical strip 0 < Re(s) < 1, and Riemann conjectured that they all lie on the critical line Re(s) = 1/2. This statement is the Riemann hypothesis, generally considered the most important open problem in pure mathematics and one of the seven Millennium Prize Problems.
The distribution of the zeros controls the distribution of the primes. The prime number theorem, which states that the number of primes up to x is asymptotically x / ln x, is equivalent to the absence of zeros on the line Re(s) = 1. Stronger bounds on the zero-free region yield stronger estimates for prime-counting error terms, and the Riemann hypothesis would give the best possible such estimates. Numerical computations have verified that the first trillions of zeros lie on the critical line, but a proof remains unknown.
Tags
complex analysis number theory prime numbers