HMMT 二月 2007 · TEAM1 赛 · 第 4 题
HMMT February 2007 — TEAM1 Round — Problem 4
题目详情
英文原题
- [ 25 ] Let F and G be two multiplicative functions, and define for positive integers n ,
( )
∑
H ( n ) = F ( d ) G .ndd | n
The number theoretic function H is called the convolution of F and G . Prove that H is multiplicative.
解析
英文解析
- [ 25 ] Let F and G be two multiplicative functions, and define for positive integers n ,
( )
∑
H ( n ) = F ( d ) G .ndd | 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
( )
∑
= F ( d ) G = H ( mn ) .mndd | mn