Following is an exercise in Rudin's Principles of Mathematical Analysis (exercise 8.21).
Let $$L_n = \frac1{2\pi}\int^\pi_{-\pi}\left|\frac{\sin(n+\frac12)x}{\sin\frac12x}\right|\,\text dx\space \space \space(n=1,2,3,...)$$
Prove that there exists a constant $C>0$ such that $$L_n>C\log n$$
or, more precisely, that the sequence $$\left\{L_n-\frac4{\pi^2}\log n\right\}$$is bounded.
I cannot do this problem because the integral is too complicated. Also, why is the last result 'more precise'?
By symmetry, we can integrate on $[0,\pi]$ and multiply by $2.$ Then let $x=2y.$ We see
$$L_n = \frac{2}{\pi}\int_0^{\pi/2}\left|\frac{\sin(2n+1)y}{\sin y}\right|\, dy.$$
Now $\sin y \le y$ on this interval. So replace $\sin y$ downstairs by $y$ and the integral gets smaller. Then make the change of variables $y = z/(2n+1).$ We obtain
$$L_n \ge \frac{2}{\pi}\int_0^{n\pi}\left|\frac{\sin z}{z}\right|\, dy \ge \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k\pi}\int_0^\pi\sin z\, dz = \frac{4}{\pi^2} \sum_{k=1}^{n}\frac{1}{k}\ge \frac{4}{\pi^2}\cdot \ln n.$$ Not sure about the "more precisely" part yet.
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Dirichlet_kernel
$$ 1+2\cos(x)+2\cos(2x)+2\cos(3x)+\cdots+2\cos(nx) = \frac{ \sin\left[\left(n+\frac{1}{2}\right)x\right\rbrack }{ \sin\left(\frac{x}{2}\right) } $$
This is a Dirichlet kernel..
$$D_n(x) = \frac{\sin(n+\frac12)x}{\sin\frac12x} = \sum\limits_{k=-n}^n e^{ikn}$$
You can complete the answer using this answer for norm of Dirichlet Kernel
Solution to the second part of the problem is as follows.
We have $L_n=4/π^2 s_n$ so that $L_n-4/π^2 \log{n}=4/π^2 (s_n-\log{n})$. By, Problem 8.9(a), $\lim_{n→∞} (s_n-\log{n})$ exists. Hence, it follows that
$$L_1>L_n-4/π^2 \log{n}>4/π^2 \lim_{n→∞} (s_n-\log{n} )$$
Hence, the sequence
$$\left\{L_n-\frac{4}{π^2} \log{n} \right\}$$
is bounded.
