HMMT 二月 2007 · TEAM1 赛 · 第 5 题

英文解析

5. [ 30 ] Prove the identity
    
   ∑ ∑2
    3
   τ ( d ) = τ ( d ) .
   d | n d | n
   Solution. Note that τ is multiplicative; in light of the convolution property just shown, it follows 3
   that both sides of the posed equality are multiplicative. Thus, it would suffice to prove the claim forkn a power of a prime. So, write n = p where p is a prime and k is a nonnegative integer. Thenk k
   ∑ ∑ ∑
   ( )
   3 i 33
   τ ( d ) = τ p = ( i + 1)
   i =0 i =0
   d | n
   ( )
   2 22
   ( k + 1) ( k + 2) ( k + 1)( k + 2)
   3 3 = 1 + · · · + ( k + 1) = =
   4 2
    
   ( )2
   k 2
   ∑ ∑
   ( )
    i = τ p = τ ( d ) ,
   i =0
   d | nas required. 2