🔢Analytic Number Theory Unit 8 – Zeta Function & Prime Number Theorem

Key Concepts and Definitions

• Analytic number theory studies arithmetic properties of integers using tools from mathematical analysis
• Zeta function $\zeta(s)$ is a complex-valued function defined as an infinite series or product over primes
• Prime number theorem describes the asymptotic distribution of prime numbers
• Riemann hypothesis conjectures that all non-trivial zeros of $\zeta(s)$ have real part $\frac{1}{2}$
• Dirichlet series $\sum_{n=1}^{\infty} \frac{a_n}{n^s}$ generalizes the zeta function
• Euler product formula expresses $\zeta(s)$ as a product over primes: $\prod_{p} (1 - p^{-s})^{-1}$
• Analytic continuation extends $\zeta(s)$ to a meromorphic function on the complex plane
• Non-trivial zeros of $\zeta(s)$ are complex numbers $\rho$ with $\zeta(\rho) = 0$ and $0 < \Re(\rho) < 1$

Historical Background

• Euler studied the zeta function in the 18th century, proving its Euler product formula
• Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude" introduced the Riemann hypothesis
• Riemann used complex analysis to study $\zeta(s)$ and relate its zeros to the distribution of primes
• Hadamard and de la Vallée Poussin independently proved the prime number theorem in 1896
  • They used complex analysis and properties of $\zeta(s)$ in their proofs
• Hardy proved in 1914 that $\zeta(s)$ has infinitely many zeros on the critical line $\Re(s) = \frac{1}{2}$
• Significant progress on the Riemann hypothesis was made in the 20th century (Lindelöf, Littlewood, Selberg, etc.)
  • However, the Riemann hypothesis remains unproven

The Riemann Zeta Function

• The Riemann zeta function is defined for $\Re(s) > 1$ by the series $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$
• It satisfies the Euler product formula $\zeta(s) = \prod_{p} (1 - p^{-s})^{-1}$ where $p$ ranges over prime numbers
• $\zeta(s)$ has a simple pole at $s=1$ with residue 1
• The functional equation relates $\zeta(s)$ and $\zeta(1-s)$: $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
• Trivial zeros of $\zeta(s)$ occur at negative even integers $s = -2, -4, -6, \ldots$
• Non-trivial zeros $\rho$ satisfy $0 < \Re(\rho) < 1$; the Riemann hypothesis states they all have $\Re(\rho) = \frac{1}{2}$
• The zero-free region $\Re(s) \geq 1$ is crucial in proving the prime number theorem

Properties and Extensions of the Zeta Function

• Analytic continuation extends $\zeta(s)$ to a meromorphic function on the complex plane
• The Dirichlet eta function $\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}$ is related to $\zeta(s)$ by $\eta(s) = (1 - 2^{1-s}) \zeta(s)$
• Dirichlet L-functions generalize $\zeta(s)$ using Dirichlet characters $\chi$: $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$
  • They satisfy similar functional equations and are used to study primes in arithmetic progressions
• The Dedekind zeta function $\zeta_K(s)$ generalizes $\zeta(s)$ to number fields $K$
• Epstein zeta functions generalize $\zeta(s)$ to quadratic forms
• The Selberg zeta function is defined using the lengths of closed geodesics on hyperbolic surfaces

Prime Number Distribution

• Let $\pi(x)$ denote the number of primes less than or equal to $x$
• The prime number theorem states that $\pi(x) \sim \frac{x}{\log x}$, i.e., $\lim_{x \to \infty} \frac{\pi(x)}{x / \log x} = 1$
  • Equivalently, the $n$-th prime number $p_n$ satisfies $p_n \sim n \log n$
• More precise estimates for $\pi(x)$ are given by the logarithmic integral $\mathrm{Li}(x) = \int_2^x \frac{dt}{\log t}$
  • The Riemann hypothesis is equivalent to $|\pi(x) - \mathrm{Li}(x)| = O(\sqrt{x} \log x)$
• Chebyshev functions $\vartheta(x) = \sum_{p \leq x} \log p$ and $\psi(x) = \sum_{p^k \leq x} \log p$ are related to $\pi(x)$
• Brun's theorem states that the sum of reciprocals of twin primes converges
• The Goldbach conjecture and twin prime conjecture are open problems related to the distribution of primes

The Prime Number Theorem

• The prime number theorem (PNT) states that $\pi(x) \sim \frac{x}{\log x}$ as $x \to \infty$
• Hadamard and de la Vallée Poussin independently proved the PNT in 1896 using complex analysis
  • Their proofs relied on the zero-free region $\Re(s) \geq 1$ for $\zeta(s)$
• Elementary proofs of the PNT were later given by Selberg and Erdős in 1949
  • These proofs avoid complex analysis and use sieve methods and tauberian theorems
• The PNT is equivalent to the statement $\sum_{p \leq x} \log p \sim x$
• The error term in the PNT is related to the zeros of $\zeta(s)$
  • The Riemann hypothesis implies the optimal error term $\pi(x) = \mathrm{Li}(x) + O(\sqrt{x} \log x)$
• The PNT has been generalized to primes in arithmetic progressions and number fields

Proof Techniques and Approaches

• Complex analysis is a key tool in analytic number theory, especially in studying $\zeta(s)$ and proving the PNT
  • Contour integration, residue theorem, and the argument principle are commonly used techniques
• Tauberian theorems relate asymptotics of series and integrals, and are used in elementary proofs of the PNT
• Sieve methods (Brun's sieve, Selberg's sieve, etc.) are used to estimate the density of sets of integers with certain properties
  • They are used in elementary proofs of the PNT and in studying the distribution of primes
• The large sieve inequality provides upper bounds for sums over characters and is used in studying primes in arithmetic progressions
• Exponential sums $\sum_{n \leq x} e^{2\pi i f(n)}$ are used to detect irregularities in the distribution of sequences
• The circle method is used to estimate the number of representations of integers by certain forms
• Modular forms and automorphic forms are used in studying L-functions and generalizations of $\zeta(s)$

Applications and Connections

• The prime number theorem has applications in cryptography, where large prime numbers are used in encryption algorithms (RSA)
• The Riemann hypothesis has implications for the distribution of prime numbers and the error term in the prime number theorem
• Analytic number theory has connections to algebraic number theory, where zeta functions of number fields are studied
• The Langlands program seeks to unify various areas of mathematics, including analytic number theory and automorphic forms
• Sieve methods have applications in computational number theory and the study of prime gaps
• Techniques from analytic number theory are used in the study of L-functions and their applications to elliptic curves and modular forms
• The Riemann zeta function has connections to physics, such as in the study of critical phenomena and quantum chaos
• Analytic number theory has applications in additive combinatorics, where techniques like exponential sums and the circle method are used