The ring of number-theoretic functions.
The ring of number-theoretic functions.
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数论函数环。
DOI：
10.2140/pjm.1959.9.975
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发表时间：
1959
影响因子：
0.6
通讯作者：
C. J. Everett
中科院分区：
文献类型：
--
作者：
E. Cashwell;C. J. Everett
导言。N={1，2，3，·}上所有函数a(N)到复数域F的集合Ω在普通加法下构成整环，其算术积定义为：(αβ)(N)=^Λa{d)β{n\d)1求和于所有d\n，dEN。该域的单位群包含所有乘法函数的集合作为一个子群。在这种背景下，数论的“逆定理”作为环运算的明显结果出现了，标准函数的推广自然地出现了。整环Ω同构于F上形式级数在可数不定集上的整环P。本文的后半部分证明了素数分解为素数的唯一分解定理在P中成立，从而在Ω中成立。
Introduction. The set Ω of all functions a(n) on N = {1, 2, 3, •} to the complex field F forms a domain of integrity under ordinary addition, and arithmetic product defined by: (α β)(n) = ^Λa{d)β{n\d)1 summed over all d \n, d e N. The group of units of this domain contains as a subgroup the set of all multiplicative functions. Against this background, the "inversion theorems" of number theory appear as obvious consequences of ring operations, and generalizations of the standard functions arise in a natural way. The domain Ω is isomorphic to the domain P of formal power series over F in a countable set of indeterminates. The latter part of the paper is devoted to proving that the theorem on unique factorization into primes, up to order and units, holds in P and hence in Ω.