For a periodic BV function $f$ which has jump discontinuties, is there any theorem in Fourier analysis which gives like $$\sum_{k=0}^n\left|c_k\right|\sim C\log\left(n\right)$$ where $C$ is a constant and $c_k$ are Fourier coefficients.

From google search, and refering to Trigonometric Series Vol.1 by Zygmund, I came to know that $$\left|c_k\right|\le \frac{V}{\pi k}$$, where $V$ is the total variation of $f$. But I could not find any theorem on the asymptotics of the summation, atleast on google.

I am making an attempt in this direction and something here