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 describes the extent to which the Dirichlet kernel behaves badly. This means that we cannot directly find a close connection between the points, but we can ensure that there are some weaker connections. In short, we do not discuss the specific content of the Lusin conjecture. What we focus on is that the Dirichlet kernel has the following control properties of the $L^1$ norm. It is especially important that when $n\rightarrow \infty$, the $L^1$ norm of $D_n$ on $[0,2\pi]$ will diverge to infinity. We can estimate:
$$
\left|D_n\right|{L^1}=\Omega(\log n) $$ by using the Riemann sum estimate in the largest neighborhood where $D_n$ is positive, and using Jensen’s inequality to handle the remaining part, it can also be proven: $$ \left|D_n\right|{L^1} \geq 4 \operatorname{Si}(\pi)+\frac{8}{\pi} \log n
$$
The exact proof of $\left|D_n\right|{L^1[0,2\pi]}=\Omega(\log n)$ is as follows:
$$
\begin{aligned}
\int_0^{2 \pi}\left|D_n(x)\right| d x & \geq \int_0^\pi \frac{|\sin [(2 n+1) x]|}{x} d x \
& \geq \sum_{k=0}^{2 n} \int_{k \pi}^{(k+1) \pi} \frac{|\sin (s)|}{s} d s \
& \geq\left|\sum_{k=0}^{2 n} \int_0^\pi \frac{\sin (s)}{(k+1) \pi} d s\right| \
& =\frac{2}{\pi} H_{2 n+1} \
& \geq \frac{2}{\pi} \log (2 n+1)
\end{aligned}
$$
Pólya–Vinogradov Inequality
Gauss sums are a special type of finite sum in algebraic number theory that involve roots on the unit circle. Gauss sums are usually denoted as:
$$
G(\chi):=G(\chi, \psi)=\sum \chi(r) \cdot \psi(r)
$$
Here, the sum is taken over elements $r$ of some finite commutative ring $R$, $\psi$ is a group homomorphism mapping the additive group $R^{+}$ to the unit circle, and $\chi$ is a group homomorphism mapping the unit group $R^x$ to the unit circle, with a value of 0 for non-unit elements $r$. Gauss sums are analogous to Gamma functions over finite fields.
Gauss sums appear everywhere in number theory. For example, in the functional equation of Dirichlet $L$-functions, for a Dirichlet character $\chi$, the equation relating $L(s, X)$ and $L(1-s, \bar{\chi})$ (where $\bar{\chi}$ is the complex conjugate of $\chi$) involves a factor:
$$
\frac{G(\chi)}{|G(\chi)|}
$$
The study of Gauss sums dates back to Carl Friedrich Gauss, who initially studied the quadratic Gauss sum, i.e., for the residue field $R$ modulo a prime $p$, and where $\chi$ is the Legendre symbol. In this case, Gauss proved that $G(\chi) = p^{1 / 2}$ or $i p^{1 / 2}$ for primes p congruent to 1 or 3 modulo 4, respectively (quadratic Gauss sums can also be computed via Fourier analysis and contour integration). Quadratic Gauss sums are closely related to the theory of theta functions. The general theory of Gauss sums developed in the early 19th century, mainly through Jacobi sums and their prime decomposition in cyclotomic fields. Gauss sums over the ring of integers modulo $N$ are linear combinations of so-called Gaussian periods. In general, the absolute value of Gauss sums is derived as an application of the Plancherel theorem over finite groups. In the case where $R$ is a field with $p$ elements and $\chi$ is nontrivial, the absolute value is $p^{1 / 2}$. Determining the exact values of general Gauss sums remains a long-standing problem. For results in certain cases, see Kummer sum.
Gauss sums have many properties, such as for a Dirichlet character modulo $N$:
$$
G(\chi)=\sum_{a=1}^N \chi(a) e^{2 x i a / N}
$$
If $\chi$ is also primitive, then:
$$
|G(\chi)|=\sqrt{N}
$$
Here are the results we have:
$$\tau(\chi) = \sum_{a=1}^p \chi(a) \cdot e\left(\frac{a}{p}\right)$$
$$\chi(n) \cdot \tau(\overline{\chi}) = \chi(n) \cdot \sum_{a=1}^p \overline{\chi}(a) \cdot e\left(\frac{a}{p}\right) = \sum_{a=1}^p \frac{\overline{\chi}(a)}{\overline{\chi}(n)} e\left(\frac{a}{p}\right) = \sum_{a=1}^p \overline{\chi}\left(\frac{a}{n}\right) \cdot e\left(\frac{a}{p}\right) = \sum_{a=1}^p \overline{\chi}(a) \cdot e\left(\frac{n a}{p}\right)$$
$$\left(\sum_{a=1}^p \overline{\chi}(a) \cdot e\left(\frac{-a}{q}\right)\right) \cdot \left(\sum_{n=1}^p \chi(a) \cdot e\left(\frac{a}{p}\right)\right) = p$$
$$\chi(n) = \frac{1}{\tau(\chi)} \cdot \sum_{n=1}^q \overline{\chi}(n) \cdot e\left(\frac{a n}{q}\right)$$
By combining these results, we obtain the Pólya–Vinogradov inequality:
The Pólya-Vinogradov inequality (1918):
There is an absolute positive constant $c$ such that for $\chi \bmod q$ non-principal,
$$
S(\chi) \leq c \sqrt{q} \log q
$$
Why the L1 norm of the Dirichlet kernel and the Pólya–Vinogradov inequality are related
Now we can clearly see that the estimation of the $L^1$ norm of the Dirichlet kernel and the Pólya-Vinogradov inequality are essentially the same thing. Fundamentally, both problems involve analyzing the behavior of a damped simple harmonic oscillation in a particular space. To better understand the connection between the two, we need to carefully examine the proof of the Pólya-Vinogradov inequality and the properties of the Dirichlet kernel.
In the proof of the Pólya-Vinogradov inequality, we manipulate $\chi(n)$ through a series of steps, revealing its connection with the $n$th characteristic function in the Dirichlet kernel. Specifically, we find that after a series of equation transformations, $\chi(n)$ exhibits a similar structure to the characteristic function in the Dirichlet kernel. However, in the context of a finite field, this connection becomes more complex, as we need to consider an additional weighting factor $\bar{\chi}(n)$. This weighting factor $\bar{\chi}(n)$ has a relatively small impact on the result, and we need to eliminate it through some simple equation transformations. After handling this weighting term, we find that both problems essentially involve estimating the behavior of a damped simple harmonic oscillation in a certain space. More specifically, we are concerned with how to describe and estimate the behavior of this damped oscillation, as well as how it exhibits logarithmic complexity as the parameter $N$ changes, which is similar to dealing with harmonic series.
By delving into the relationship between these two problems, we find that they both rely on accurately analyzing and estimating some kind of damped simple harmonic oscillation. This connection provides us with a unique perspective, enabling us to understand the estimation of the $L^1$ norm of the Dirichlet kernel, the Pólya-Vinogradov inequality, and their relationship from a broader mathematical background.