Although the $L^1$ norm of the Dirichlet kernel and the Pólya-Vinogradov inequality belong to different areas of mathematics, they share some connections. L1 norm of the Dirichlet kernel $$D_n(x)=\sum_{k=-n}^n e^{i k x}=\left(1+2 \sum_{k=1}^n \cos (k x)\right)=\frac{\sin ((n+1 / 2) x)}{\sin (x / 2)}$$ The Dirichlet kernel is mainly used to describe the pointwise relationship between a function f and its Fourier transform hat f. We have a famous conjecture, the Lusin conjecture, which roughly de…
