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The Riemann zeta function or Euler–Riemann zeta function, ''ζ''(''s''), is a function of a complex variable ''s'' that analytically continues the sum of the infinite series : which converges when the real part of ''s'' is greater than 1. More general representations of ''ζ''(''s'') 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.〔This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be 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 Apéry proved the irrationality of ''ζ''(3). 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''. (The notation ''s'', ''σ'', and ''t'' is used traditionally in the study of the ''ζ''-function, following Riemann.) The following infinite series converges for all complex numbers ''s'' with real part greater than 1, and defines ''ζ''(''s'') in this case: : It can also be defined by the integral : where Γ(''s'') is the gamma function. The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series. Leonhard Euler considered the above series in 1740 for positive integer values of ''s'', and later Chebyshev extended the definition to real ''s'' > 1. The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for ''s'' such that 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 . For ''s'' = 1 the series is the harmonic series which diverges to +∞, and : Thus the Riemann zeta function is a meromorphic function on the whole complex ''s''-plane, which is holomorphic everywhere except for a simple pole at ''s'' = 1 with residue 1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemann zeta function」の詳細全文を読む スポンサード リンク
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