Reading this text from Wikipedia, I found the following statement about the Dirichlet kernel:
$$\| D_n \|_{L^1} \approx \log n, $$
where $\approx$ denotes "is of the order". I think that this mean that
$$\displaystyle{\lim_{n\to\infty} \frac{\|D_n\|_{L^1}}{\log n} = 1},$$
as usual. I have trying to prove this, but without success. Someone can to say me how to show this? Or maybe, indicate to me a text where I can find a proof of this?
Thanks!
Note 1: $D_n$ is the Dirichlet kernel, i.e., $$ D_n(t) := \sum_{k=-n}^{n} e^{i kt} = 1 + 2\sum_{k=1}^{n} \cos kt. $$
Note 2: $\|D_n\|_{L^1}$ is the $L^1$ norm of $D_n$, i.e., $$\displaystyle{\|D_n\|_{L^1} := \int_{-\pi}^{\pi} |D_n(t)|dt }.$$