I guess that the following estimates are classical. Let $D_N$ be the $1D$ Dirichlet kernel, $$ D_N(t)=\frac{\sin((N+\frac12)t)}{\sin (t/2)}. $$ We have for $1<p<\infty$, \begin{align} \Vert D_N\Vert_{L^p(0,2π)}&\sim_{N\rightarrow +\infty} N^{1-\frac 1p} \frac{2}{π^{1/p}} \left( \int_{0}^{+\infty} {\frac{\vert\sin t\vert^p}{t^p} }dt\right)^{1/p}, \end{align} and $ \Vert D_N\Vert_{L^1(0,2π)}\sim_{N\rightarrow +\infty}\frac{4 }{π^2}\ln N. $ Of course, here $a_N\sim b_N$ means $$ a_N=b_N(1+\epsilon _N), \quad \lim \epsilon_N=0. $$ Although I found in the literature some sort of equivalence, say the $L^1$ norm is between $c_1\ln N$ and $c_2\ln N$, the above form is not so easy to locate.

My question: I would be interested in a full expansion, using some type of Euler-Maclaurin formula, say for the $L^p$ norm of the Dirichlet kernel, with $$ \Vert D_N\Vert_{L^p(0,2π)}= c_0 N^{1-\frac1p}+c_1 N^{-\frac1p}+O(N^{-1-\frac1p}). $$ Is it written anywhere?