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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 Explicit formulae (L-function) ： ウィキペディア英語版 Explicit formulae (L-function) In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field. ==Riemann's explicit formula== In his 1859 paper ''On the Number of Primes Less Than a Given Magnitude'' Riemann found an explicit formula for the normalized prime-counting function π0(''x'') which is related to the prime-counting function π(''x'') by :$\backslash pi\_0(x)\; =\; \backslash frac\; \backslash lim\_\; (\backslash pi(x+h)\; +\; \backslash pi(x-h)).$ His formula was given in terms of the related function :$f(x)\; =\backslash pi(x)+\backslash frac\backslash pi(x^)+\backslash frac\backslash pi(x^)+\backslash cdots$ which counts primes where a prime power ''p''''n'' counts as 1/''n'' of a prime and which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities. The normalized prime-counting function can be recovered from this function by :$\backslash pi\_0(x)\; =\; \backslash sum\_n\backslash mu(n)f(x^)/n\; =\; f(x)\; -\backslash fracf(x^)-\backslash fracf(x^)\; -\; \backslash cdots.$ Riemann's formula is then :$f(x)\; =\; \backslash operatorname(x)\; -\; \backslash sum\_\backslash rho\; \backslash operatorname(x^\backslash rho)\; -\backslash log(2)\; +\backslash int\_x^\backslash infty\backslash frac$ involving a sum over the non-trivial zeros ρ of the Riemann zeta function. The sum is not absolutely convergent, but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral :$\backslash operatorname(x)\; =\; \backslash int\_0^x\backslash frac.$ The terms li(''x''ρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined by analytic continuation in the complex variable ρ in the region ''x''>1 and Re(ρ)>0. The other terms also correspond to zeros: the dominant term li(''x'') comes from the pole at ''s'' = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see .) A simpler variation of Riemann's formula using the normalization $\backslash psi\_0$ of Chebyshev's function ψ rather than π is〔Weisstein, Eric W. ( Explicit Formula ) on MathWorld.〕 von-Mangoldt's explicit formula :$\backslash psi\_0(x)\; =\; \backslash dfrac\backslash int\_0^\backslash left(-\backslash dfrac\backslash right)\backslash dfracds=x-\backslash sum\_\backslash rho\backslash frac\; -\; \backslash log(2\backslash pi)\; -\backslash log(1-x^)/2$ where for non-integral ''x'', ψ(''x'') is the sum of log(''p'') over all prime powers ''p''''n'' less than ''x''. It plays an important role in von Mangoldt's proof of Riemann's explicit formula. Here the sum over zeroes should again be taken in increasing order of imaginary part:〔Ingham (1990) p.77〕 :$\backslash sum\_\backslash rho\backslash frac\; =\; \backslash lim\_\; S(x,T)\; \backslash$ where :$S(x,T)\; =\; \backslash sum\_\; \backslash frac\; \backslash \; .$ The error involved in truncating the sum to ''S''(''x'',''T'') is of order〔 :$x^2\; \backslash frac\; +\; \backslash log\; x\; \backslash \; .$ 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「Explicit formulae (L-function)」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.