# Analytic number theory
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\chapter{L-functions}

It's often somewhat surprising to discover that mathematical analysis has applications in number theory.


The main 'subfields' or 'applications' of analytic number theory are for \emph{additive number theory and multiplicative number theory}.

Additive number theory uses analysis to study integer subsets under addition (like Goldbach conjecture, Waring's problem etc.). The main objects of interest are sumsets of subsets of Abelian groups (most often $\mathbb{Z}$, given that this is number theory).

Multiplicative number theory uses analysis to propound propositions/theorems about , usually by applying analysis to arithmetic functions.



\[ \prod_{p} (\sum_{n=1}^{\infty} \frac{1}{p^n} ) = \prod_{p} (\frac{1}{ 1- p^{-1}} ) = \sum_{n=1}^{\infty} \frac{1}{n} \]

\begin{definition}[Riemann $\zeta$ function]
\[ \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \]
\end{definition}
\[ \zeta (s) = \prod_{p | \text{ prime}} \frac{1}{1-p^{-s}} \]
\[ \zeta (s) = \frac{1}{\Gamma (s)} \int_{0}^{\infty} \frac{x^{s-1}}{e^x -1}dx \]
\[ \zeta (2) = \frac{\pi^2}{6} \]


\begin{definition}[Dirichlet $\eta$ function]
\[ \eta (s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} \]
\end{definition}







\chapter{$p$-adic analysis}


\begin{definition}[Prime-counting function]
The \emph{prime-counting function} is the function $\pi : \mathbb{R}_{+} \to \mathbb{N}$ that enumerates the amount of primes less than its argument.
\[ \pi(x) = | \{ p \in [1,x] : p \text { is prime } \}| \]
\end{definition}


\begin{definition}[First Chebyshev function]
The \emph{first Chebyshev function} is the function $\vartheta : \mathbb{R}_{+} \to \mathbb{N}$ defined as such.
\[ \pi(x) = | \{ \ln (p) : p \text { is prime } \land p \in [1,x] \}| \]
\end{definition}

\begin{proposition}
\[ \pi (x) \geq \lfloor \log_2 (\log_2 (x)) \rfloor +1 \]
\end{proposition}


\begin{theorem}{Prime number theorem]
\[\pi (x) \sim \frac{x}{\ln (x)}\]
\[ \pi (x)  \sim \mathrm{Li} (x) \]
\end{theorem}














\chapter{Stuff to sort}
\section{Lambert function}