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March 17th, 2019, 08:56 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 Thanks: 2  Dirichlet Convolution question
if p is prime and k>1 then let $f(p^{k})=log(p)$, for all other n let f(n)=0. Prove that$(f*u)(n)=log(n)$ for all n where u(n)=1 for all n.  Ok so for this question i have started by saying \[(f*u)(n)=\sum_{jn}f(j)*u(\frac{n}{j})=f(1)*u(p^k) +f(p)*u(p^{k1})+...f(p^k)*u(1)\] but i don't know how i can go from here since i can not see how i get to the end answer. I was thinking maybe i could various values of d which divide n expressed in the prime factorisation?? Thanks to anyone who can solve this. Last edited by Jaket1; March 17th, 2019 at 09:03 AM. 
March 23rd, 2019, 01:34 PM  #2 
Newbie Joined: Sep 2017 From: Belgium Posts: 19 Thanks: 7 
Did you meant "and $k\geqslant1$"? with Von Mangoldt $\Lambda * 1 = log$ Well, if you look at $n=p_{i_1}^{j_1}\cdot p_{i_2}^{j_2}....$ its prime factorization, and knowing $log(ab)=log(a)+log(b)$ or $log(a^b)=b\cdot log(a)$, you should be fine with sums of logs. 

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convolution, dirichlet, question 
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