HMMT 二月 2007 · TEAM1 赛 · 第 4 题
HMMT February 2007 — TEAM1 Round — Problem 4
题目详情
- [ 25 ] Let F and G be two multiplicative functions, and define for positive integers n , ( ) ∑ n H ( n ) = F ( d ) G . d d | n The number theoretic function H is called the convolution of F and G . Prove that H is multiplicative.
解析
- [ 25 ] Let F and G be two multiplicative functions, and define for positive integers n , ( ) ∑ n H ( n ) = F ( d ) G . d d | n The number theoretic function H is called the convolution of F and G . Prove that H is multiplicative. Solution. Let m and n be relatively prime positive integers. We have ( ) ( ) ∑ ∑ m n ′ H ( m ) H ( n ) = F ( d ) G F ( d ) G ′ d d ′ d | m d | n ( ) ( ) ( ) ∑ ∑ m n mn ′ ′ = F ( d ) F ( d ) G G = F ( dd ) G ′ ′ d d dd ′ ′ d | m, d | n d | m, d | n ( ) ∑ mn = F ( d ) G = H ( mn ) . d d | mn