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.


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 $a=5$, $b=11$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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