My Math Forum (http://mymathforum.com/math-forums.php)
-   Number Theory (http://mymathforum.com/number-theory/)
-   -   Dirichlet Convolution question (http://mymathforum.com/number-theory/345972-dirichlet-convolution-question.html)

 Jaket1 March 17th, 2019 08:56 AM

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

$(f*u)(n)=\sum_{j|n}f(j)*u(\frac{n}{j})=f(1)*u(p^k) +f(p)*u(p^{k-1})+...f(p^k)*u(1)$

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.

 Collag3n March 23rd, 2019 01:34 PM

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.

 All times are GMT -8. The time now is 01:14 PM.