section_path: Analytic properties/Zeta function and the Riemann hypothesis
section_name: Zeta function and the Riemann hypothesis
paragraph_text: Zeta function and the Riemann hypothesis
  
  One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the Riemann zeta function ζ (s) {\displaystyle \zeta (s)} are located.This function is an analytic function on the complex numbers. For complex numbers s {\displaystyle s} with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers,
  
  This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another proof that there are infinitely many primes: if there were only finitely many,then the sum-product equality would also be valid at s = 1 {\displaystyle s=1}, but the sum would diverge (it is the harmonic series 1 + 1 2 + 1 3 + … {\displaystyle 1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\dots }) while the product would be finite, a contradiction.
  
  The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the prime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1, although other more elementary proofs have been found. The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of x {\displaystyle x} for intervals near a number x {\displaystyle x}).