What are the asymptotic bounds for the $L^1$-norm of the Dirichlet kernel?

Solution 1:

$$f(x)=\frac1{|\sin(x/2)|}-\frac{1}{|x/2|}\in C^0[-\pi,\pi]$$ So the asymptotic follows easily $$\int_{-\pi}^\pi \left|\frac{\sin((n+1/2)x)}{\sin(x/2)}\right|dx=\int_{-\pi}^\pi |\sin((n+1/2)x)| f(x)dx+\int_{-\pi}^\pi \left|\frac{\sin((n+1/2)x)}{x/2}\right|dx $$ $$ =\int_{-\pi}^\pi \frac{2}\pi f(x)dx+o(1)+4\int_0^{(n+1/2)\pi} \left|\frac{\sin(x)}{x}\right|dx $$ $$ = C+o(1)+4\int_1^{(n+1/2)\pi} \frac{2/\pi}{|x|}dx=C+\frac8\pi\log \pi +o(1)+\frac8\pi\log n$$ where $$C=\int_{-\pi}^\pi \frac{2}\pi f(x)dx+ 4\int_0^1 \frac{|\sin(x)|}{|x|}dx+4\int_1^\infty\frac{|\sin(x)|-2/\pi}{|x|}dx$$

I think the $o(1)$ term can be improved to $\sum_{j=1}^J a_j n^{-j}+O(n^{-J-1})$