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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 Abstract analytic number theory ： ウィキペディア英語版 Abstract analytic number theory Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results. The theory was invented and developed by mathematicians such as John Knopfmacher and Arne Beurling in the twentieth century. == Arithmetic semigroups == The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid ''G'' satisfying the following properties: *There exists a countable subset (finite or countably infinite) ''P'' of ''G'', such that every element ''a'' ≠ 1 in ''G'' has a unique factorisation of the form ::$a\; =\; p\_1^\; p\_2^\; \backslash cdots\; p\_r^$ :where the ''p''''i'' are distinct elements of ''P'', the α''i'' are positive integers, ''r'' may depend on ''a'', and two factorisations are considered the same if they differ only by the order of the factors indicated. The elements of ''P'' are called the ''primes'' of ''G''. *There exists a real-valued ''norm mapping'' $|\backslash mbox|$ on ''G'' such that *#$|1|\; =\; 1$ *#$|p|\; >\; 1\; \backslash mbox\; p\; \backslash in\; P$ *#$|ab|\; =\; |a|\; |b|\; \backslash mbox\; a,b\; \backslash in\; G$ *#The total number $N\_G(x)$ of elements $a\; \backslash in\; G$ of norm $|a|\; \backslash leq\; x$ is finite, for each real $x\; >\; 0$. 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「Abstract analytic number theory」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.