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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Riemann hypothesis ： ウィキペディア英語版 Riemann hypothesis In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and the complex numbers with real part 1/2. It was proposed by , after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics . The Riemann hypothesis, along with Goldbach's conjecture, is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems. The Riemann zeta function ζ(''s'') is a function whose argument ''s'' may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(''s'') = 0 when ''s'' is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called ''non-trivial zeros''. The Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that: :The real part of every non-trivial zero of the Riemann zeta function is . Thus, if the hypothesis is correct, all the non-trivial zeros lie on the critical line consisting of the complex numbers where ''t'' is a real number and ''i'' is the imaginary unit. There are several nontechnical books on the Riemann hypothesis, such as , , , . The books , , and give mathematical introductions, while , and are advanced monographs. ==Riemann zeta function== The Riemann zeta function is defined for complex ''s'' with real part greater than 1 by the absolutely convergent infinite series :$\backslash zeta(s)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$ Leonhard Euler showed that this series equals the Euler product :$\backslash zeta(s)\; =\; \backslash prod\_\backslash right)\backslash zeta(s)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash cdots.$ However, the series on the right converges not just when ''s'' is greater than one, but more generally whenever ''s'' has positive real part. Thus, this alternative series extends the zeta function from to the larger domain , excluding the zeros $s\; =\; 1\; +\; 2\backslash pi\; in/\backslash ln(2)$ of $1-2/2^s$ (see Dirichlet eta function). The zeta function can be extended to these values, as well, by taking limits, giving a finite value for all values of ''s'' with positive real part except for a simple pole at ''s'' = 1. In the strip the zeta function satisfies the functional equation :$\backslash zeta(s)\; =\; 2^s\backslash pi^\backslash \; \backslash sin\backslash left(\backslash frac\backslash right)\backslash \; \backslash Gamma(1-s)\backslash \; \backslash zeta(1-s).$ One may then define ζ(''s'') for all remaining nonzero complex numbers ''s'' by assuming that this equation holds outside the strip as well, and letting ζ(''s'') equal the right-hand side of the equation whenever ''s'' has non-positive real part. If ''s'' is a negative even integer then ζ(''s'') = 0 because the factor sin(π''s''/2) vanishes; these are the trivial zeros of the zeta function. (If ''s'' is a positive even integer this argument does not apply because the zeros of the sine function are cancelled by the poles of the gamma function as it takes negative integer arguments.) The value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(''s'') as ''s'' approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all non-trivial zeros lie in the critical strip where ''s'' has real part between 0 and 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 hypothesis」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.