This course develops the analytic side of number theory through the study of multiplicative functions, Dirichlet series, and the analytic behavior of zeta and $L$-functions. It begins with arithmetic functions and Dirichlet convolution as the algebraic language for encoding number-theoretic data, then turns to Dirichlet series as generating functions that translate sums over integers into complex-analytic objects. From there, the course focuses on the Riemann zeta function near $\operatorname{Re}(s)>1$, Perron's formula, and summatory-function techniques that connect analytic estimates back to counting problems.

A central theme is how analytic information about generating functions controls the distribution of primes. After proving the prime number theorem for $\zeta(s)$, the course introduces Dirichlet characters and orthogonality to study arithmetic progressions, leading to Dirichlet $L$-functions and their Euler products. The later chapters establish nonvanishing at $s=1$, prove Dirichlet's theorem on primes in arithmetic progressions, and extend the framework to completed $L$-functions and functional equations. Each stage builds on the previous one: algebraic structure leads to analytic continuation, analytic continuation leads to asymptotics, and asymptotics reveal deep prime-distribution phenomena.

# Introduction

This course asks how analytic objects remember arithmetic information. We begin with functions on the positive integers, package them into Dirichlet series, and study the resulting complex functions near their singularities. The guiding example is the passage from the zeta function to the prime number theorem, followed by the extension from all primes to primes in arithmetic progressions through Dirichlet characters and $L$-functions.

The course assumes elementary number theory, complex analysis, real analysis, and finite abelian groups. The first part builds the algebra of arithmetic functions and Dirichlet series. The second part uses complex-analytic continuation, residues, zero-free regions, and Tauberian reasoning to extract asymptotic information about primes.

## The Central Problem

The first question is how to turn statements about divisibility and primes into analytic statements about functions of a complex variable. Counting primes directly is difficult because primality is not additive or smooth. Analytic number theory replaces the primes by weighted sums and generating functions whose singularities encode growth.

[definition: Prime Counting Functions] Let $x \ge 1$. The prime counting functions used in this course are \begin{align*} \pi(x) &:= |\{p \le x : p \text{ prime}\}|,\\ \vartheta(x) &:= \sum_{p \le x} \log p,\\ \psi(x) &:= \sum_{n \le x} \Lambda(n), \end{align*} where $\Lambda$ is the von Mangoldt function. [/definition]

The function $\pi(x)$ is the most direct object, but $\vartheta(x)$ and $\psi(x)$ are better adapted to products over primes and logarithmic differentiation. A major theme of the course is that these three functions contain essentially the same first-order information.

To make $\psi(x)$ usable, we need the weight that counts prime powers with exactly the logarithmic mass contributed by their underlying prime. This is the arithmetic function that will later appear when Euler products are logarithmically differentiated.

[definition: Von Mangoldt Function] The von Mangoldt function is the arithmetic function $\Lambda: \mathbb N \to \mathbb R$ defined by \begin{align*} \Lambda(n) := \begin{cases} \log p, & n = p^k \text{ for some prime } p \text{ and } k \in \mathbb N,\\ 0, & \text{otherwise}. \end{cases} \end{align*} [/definition]

The weight $\Lambda(n)$ treats prime powers as the natural logarithmic atoms of the integers. It appears when one differentiates Euler products, so it forms the bridge between prime factorisation and analytic functions.

The prime number theorem will be the endpoint of the zeta-function part of the course, but the objects needed for its proof appear earlier. The function $\Lambda$ is the version of prime counting that enters the logarithmic derivative of an Euler product, so the path begins by treating arithmetic functions as algebraic data before introducing their Dirichlet series.

## Arithmetic Functions as Data

The next question is what kind of arithmetic data should be studied before introducing complex variables. An arithmetic function is a sequence indexed by the positive integers, but multiplication of integers suggests a convolution product rather than pointwise multiplication.

[definition: Arithmetic Function] An arithmetic function is a function $f: \mathbb N \to \mathbb C$. [/definition]

We use the standard notation $\tau(n)$ for the number of positive divisors of $n$, $\sigma_k(n):=\sum_{d\mid n}d^k$ for the $k$th divisor-sum function, $\varphi(n):=|\{1\le a\le n:\gcd(a,n)=1\}|$ for Euler's totient function, and $\mu(n)$ for the Mobius function.

Arithmetic functions form a vector space under pointwise addition and scalar multiplication. Pointwise multiplication misses the way an integer is assembled from its divisors: multiplying $f(n)$ and $g(n)$ only compares the two functions at the same integer $n$, while divisor identities need all decompositions $n=ab$. The useful multiplication for this course is Dirichlet convolution, because divisors are the natural finite pieces of the multiplicative structure of $\mathbb N$.

[definition: Dirichlet Convolution] For arithmetic functions $f,g: \mathbb N \to \mathbb C$, their Dirichlet convolution is the arithmetic function $f*g: \mathbb N \to \mathbb C$ defined by \begin{align*} (f*g)(n) := \sum_{d \mid n} f(d)g(n/d). \end{align*} [/definition]

The definition is deliberately compressed, so the first useful check is whether it recovers the familiar divisor-counting function.