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I don't know much about number theory, it seems this summation might involve some facts from number theory.

Could you give me some idea of doing it? Thank you very much. The summation is

$$ g(n)=\sum_{d|n} f(d) $$

where $f(d)=k^d$, for some fixed positive integers $k$ and $n$.

for example,

if $n=6$, $g(6)=k+k^2+k^3+k^6$. thanks

if no formula, what is the asymptomatic behaviour of it when n large. thanks again.

share|improve this question
    
It's annoying that $g$ is not multiplicative. One obvious observation: $g(n) = \Theta(k^n).$ –  user7530 Jan 2 '13 at 9:35
    
Well let's see, it's the Dirichlet product of $f$ with the all $1$'s function. I don't know if that helps though. –  Alexander Gruber Jan 3 '13 at 0:10
    
$$\sum_{d\mid n}x^d=\sum_{k=1}^\infty\frac{ln(x)^k\sigma_k(n)}{k!}$$ –  Ethan Feb 2 '13 at 4:48

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