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.



An asymptotic formula for this sum
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 $a=5$, $b=11$.


![](https://ilorentz.org/beenakker/MO/largexsum.png)


