A formula to calculate summations over all divisors of a fixed integer

Asked Jan 02 '13 at 08:04

Active Jan 02 '13 at 08:53

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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.

edited Jan 02 '13 at 08:53

asked Jan 02 '13 at 08:04

Frank W

It's annoying that $g$ is not multiplicative. One obvious observation: $g(n) = \Theta(k^n).$ – user7530 Jan 02 '13 at 09:35
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 03 '13 at 00:10
$$\sum_{d\mid n}x^d=\sum_{k=1}^\infty\frac{ln(x)^k\sigma_k(n)}{k!}$$ – Ethan Splaver Feb 02 '13 at 04:48

0 Answers0