What is the complexity of sum of Logarithm of an arithmetic series
$f(a,b)=\sum_{k=1}^b \log(k*\frac{a}{b})=\log\prod_{k=1}^b \left(k*\frac{a}{b}\right)=\log \frac{b!*a^b}{b^b}=\log(b!)+b\log\frac{a}{b}=b\log a +[\log(b!)-b\log(b)]$

by Stirling's formula, $\log(b!)=b\log b -b +\mathcal{O}(\log{b})$

Therefore $f(a,b)=b\log a +o(b\log a)$ so $f(a,b)\sim b\log a$

NB: all asymptotics ($\sim$, $\mathcal{O}$ and $o$) are taken as $a,b\to\infty$ with $a&lt;b$)

For a more precise evaluation than just an equivalent, replace Stirling's approximation (https://en.wikipedia.org/wiki/Stirling%27s_approximation) with Stirling's bounds (same wikipedia article).