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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 Vinogradov's mean-value theorem ： ウィキペディア英語版 Vinogradov's mean-value theorem Vinogradov's mean value theorem is an upper bound for $J\_(X)$, the number of solutions to the system of $k$ simultaneous Diophantine equations in $2s$ variables given by $x\_1^j+x\_2^j+\backslash cdots+x\_s^j=y\_1^j+y\_2^j+\backslash cdots+y\_s^j\backslash quad\; (1\backslash le\; j\backslash le\; k)$ with $1\backslash le\; x\_i,y\_i\backslash le\; X,\; (1\backslash le\; i\backslash le\; s)$. An analytic expression for $J\_(X)$ is $J\_(X)=\backslash int\_|f\_k(\backslash mathbf\backslash alpha;X)|^d\backslash mathbf\backslash alpha$ where $f\_k(\backslash mathbf\backslash alpha;X)=\backslash sum\_\backslash exp(2\backslash pi\; i(\backslash alpha\_1x+\backslash cdots+\backslash alpha\_kx^k)).$ A strong estimate for $J\_(X)$ is an important part of the Hardy-Littlewood method for attacking Waring's problem and also for demonstrating a zero free region for the Riemann zeta-function in the critical strip.〔E. C. Titchmarsh (rev. D. R. Heath-Brown): The theory of the Riemann Zeta-function, OUP〕 Various bounds have been produced for $J\_(X)$, valid for different relative ranges of $s$ and $k$. The classical form of the theorem applies when $s$ is very large in terms of $k$. == The conjectured form == By considering the $X^s$ solutions where $x\_i=y\_i,\; (1\backslash le\; i\backslash le\; s)$ we can see that $J\_(X)\backslash gg\; X^s$. A more careful analysis (see Vaughan 〔R.C. Vaughan: The Hardy-Littlewood method, CUP〕 equation 7.4) provides the lower bound $J\_\backslash gg\; X^s+X^.$ The main conjectural form of Vinogradov's mean value theorem is that the upper bound is close to this lower bound. More specifically that for any $\backslash epsilon>0$ we have $J\_(X)\backslash ll\; X^+X^.$ If $s\backslash ge\; k(k+1)$ this is equivalent to the bound $J\_(X)\backslash ll\; X^.$ Similarly if $s\backslash le\; k(k+1)$ the conjectural form is equivalent to the bound $J\_(X)\backslash ll\; X^.$ Stronger forms of the theorem lead to an asymptotic expression for $J\_$, in particular for large $s$ relative to $k$ the expression $J\_\backslash sim\; \backslash mathcal\; C(s,k)X^,$ where $\backslash mathcal\; C(s,k)$ is a fixed positive number depending on at most $s$ and $k$, holds. 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「Vinogradov's mean-value theorem」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.