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

For $$n \in \mathbb{N}$$, the Dirichlet kernel $$D_n$$ is the function on the circle $$\mathbb{T} = \mathbb{R}/\mathbb{2\pi \mathbb{Z}}$$ defined by $$D_n(x) = \frac{ \sin [(n+1/2)x] }{ \sin(x/2) } \quad \text{for } x \in \mathbb{T}$$ where each $$x \in \mathbb{T}$$ is identified with a point in the interval $$(-\pi, \pi]$$. The Dirichlet kernel is important in Fourier analysis because the Fourier partial sum can be given as $$S_nf = f * D_n$$ for $$f \in L^1(\mathbb{T})$$ and $$n \in \mathbb{N}$$.

The $$L^1$$-norm of the Dirichlet kernel is given by

$$||D_n||_1 = \frac{1}{2\pi} \int_{-\pi}^\pi \left| \frac{ \sin [(n+1/2)x] }{ \sin(x/2) } \right| dx$$

The factor $$1/2\pi$$ is here to normalize the length of $$\mathbb{T}$$ to $$1$$.

With this definition of $$L^1$$-norm, it is known that

$$||D_n||_1 = \frac{4}{\pi^2} \ln n + O(1)$$

Thus, the limsup and liminf

$$\overline{C} = \limsup_{n \to \infty} \left( ||D_n||_1 - \frac{4}{\pi^2} \ln n \right) \quad \text{and} \quad \underline{C} = \liminf_{n \to \infty} \left( ||D_n||_1 - \frac{4}{\pi^2} \ln n \right)$$

exist and are finite.

My question is, are the exact values of $$\overline{C}$$ and $$\underline{C}$$ known? If not, what are the current best known upper and lower bounds for $$\overline{C}$$ and $$\underline{C}$$?

I know that

$$||D_n||_1 \geq \frac{4}{\pi^2} \ln n + \frac{2}{\pi} \int_0^\pi \frac{\sin t}{t} dt \quad \text{for all } n \in \mathbb{N}$$

Thus, a trivial lower bound is

$$\overline{C} \geq \underline{C} \geq \frac{2}{\pi} \int_0^\pi \frac{\sin t}{t} dt$$

$$f(x)=\frac1{|\sin(x/2)|}-\frac{1}{|x/2|}\in C^0[-\pi,\pi]$$ 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 $$\sum_{j=1}^J a_j n^{-j}+O(n^{-J-1})$$