Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable defined as ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 1 s + 1 2 s + 1 3 s + ⋯ {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots } for R e ( s ) > 1 {\displaystyle \mathrm {Re} (s)>1} , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. 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. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics. The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979, Roger Apéry proved the irrationality of ζ(3), and as a consequence, the number was named after him. The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

Definition

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re(s) = σ > 1, the function can be written as a converging summation or as an integral:

ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 Γ ( s ) ∫ 0 ∞ x s − 1 e x − 1 d x , {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\,,}

where

Γ ( s ) = ∫ 0 ∞ x s − 1 e − x d x {\displaystyle \Gamma (s)=\int _{0}^{\infty }x^{s-1}\,e^{-x}\,\mathrm {d} x}

is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1. Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to Re(s) > 1. The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1, the series is the harmonic series which diverges to +∞, and

lim s → 1 ( s − 1 ) ζ ( s ) = 1. {\displaystyle \lim _{s\to 1}(s-1)\zeta (s)=1.}

Thus the Riemann zeta function is a meromorphic function on the whole complex plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

Euler's product formula In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

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