page_title: Riemann zeta function
section_title: Zeros, the critical line, and the Riemann hypothesis
hierarchical_section_title: Riemann zeta function / Zeros, the critical line, and the Riemann hypothesis
context_page_description: The Riemann zeta function or Euler–Riemann zeta function, ζ, is a function of a complex variable s that analytically continues the sum of the Dirichlet series
  which converges when the real part of s is greater than 1. More general representations of ζ for all s are given below. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
  As a function of a real variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis, which was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.
  The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ, provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ.
context_section_description: The functional equation shows that the Riemann zeta function has zeros at −2, −4,…. These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin  being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s ∈ ℂ : 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = . In the theory of the Riemann zeta function, the set {s ∈ ℂ : Re(s) = } is called the critical line. For the Riemann zeta function on the critical line, see Z-function.
caption: Apart from the trivial zeros, the Riemann zeta function has no zeros to the right of σ = 1 and to the left of σ = 0 (neither can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = 1/2 and, according to the Riemann hypothesis, they all lie on the line σ = 1/2.