数论函数中的R－空间的一个性质

A PROPERTY OF R-SPACE IN ARITHMETIC FUNCTIONS

本文证明了如果Ｔ是Ｒ－空间，ｆ是一个完全积性函数，若有ｋ次复系数非零多项式Ｐ（ｚ）使Ｐ（Ｅ）ｆ∈Ｔ，则一定有正整数Ｂ使（ＥＢ－１）ｋｆ∈Ｔ，Ｂ的所有素因子ｐ满足ｆ（ｐ）＝０，并且如果Ｐ（ｚ）是满足条件的最低次首１多项式，则Ｐ（ｚ）一定可以表示为若干个分圆多项式Ｆｄ（ｚ）的来积．此外使ｆ（ｐ）＝Ｏ的素数ｐ的个数不超过ｋ－１．

We proved that: Let T be a R -space,and f be a completely multiplicative function,if P(E)f∈ T holds for some polynomial p(z)∈ C[z],then there exists a suitable integer B≥ 1 such that (EB-1 )kf∈ T, B contains only such prime factors p for which f(p)=0 and k≥1 is the degree of p(z). Furthermore,if p(z) is a minimal degree polynomialwhich satisfies above conditions, then p(z) must be a product of some cyclotomic polynomials,and there exists at most k - 1 primes q for which f(q) = 0.