HMMT 二月 2007 · TEAM1 赛 · 第 5 题

HMMT February 2007 — TEAM1 Round — Problem 5

解析

[ 30 ] Prove the identity   2 ∑ ∑ 3   τ ( d ) = τ ( d ) . d | n d | n 3 Solution. Note that τ is multiplicative; in light of the convolution property just shown, it follows that both sides of the posed equality are multiplicative. Thus, it would suffice to prove the claim for k n a power of a prime. So, write n = p where p is a prime and k is a nonnegative integer. Then k k ∑ ∑ ∑ ( ) 3 3 i 3 τ ( d ) = τ p = ( i + 1) i =0 i =0 d | n ( ) 2 2 2 ( k + 1) ( k + 2) ( k + 1)( k + 2) 3 3 = 1 + · · · + ( k + 1) = = 4 2   2 ( ) 2 k ∑ ∑ ( ) i   = τ p = τ ( d ) , i =0 d | n as required. 2