5
$\begingroup$

Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.

$$\sum_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n},$$ where $a$ and $b$ are two coprime integers. Thanks in advance.

$\endgroup$
9
$\begingroup$

The sum $$F(X)=\sum_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n}=\sum_{p={\rm Int}\,[-a/b]}^{{\rm Int}\,[(x-a)/b]}\log(a+pb)$$ can be approximated in the large-$X$ limit by $$F_\infty(X)=\sum_{p=1}^{(X-a)/b}\log(pb)=\frac{X-a}{b}\log b+\log\Gamma\left(\frac{X-a}{b}+1\right)$$

Here is a plot of $F(X)$ (gold) and $F_\infty(X)$ (blue) for $a=5$, $b=11$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.