🔢Algebraic Number Theory Unit 14 – Zeta Functions and L–Functions

Introduction to Zeta and L-Functions

• Zeta functions and L-functions are complex-valued functions that play a central role in analytic number theory
• Serve as generating functions for various arithmetic objects (prime numbers, algebraic numbers, elliptic curves)
• Provide a powerful tool for studying the distribution of prime numbers and other arithmetic properties
• Exhibit deep connections to complex analysis, algebra, and geometry
• Have applications in cryptography, coding theory, and physics (quantum field theory, string theory)
• Offer insights into the behavior of arithmetic functions and the structure of number fields
• Generalize the concept of the Riemann zeta function to a broader class of functions

Historical Background and Motivation

• Leonhard Euler introduced the zeta function in the 18th century while studying the distribution of prime numbers
  • Euler's zeta function: $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\Re(s) > 1$
• Bernhard Riemann extended Euler's work and established the connection between the zeta function and prime numbers in his groundbreaking 1859 paper
• Riemann's work laid the foundation for the study of L-functions and their role in number theory
• Motivation for studying zeta and L-functions stems from their ability to encode arithmetic information
  • Riemann Hypothesis: non-trivial zeros of the Riemann zeta function lie on the critical line $\Re(s) = \frac{1}{2}$
• Generalizations of the Riemann zeta function led to the development of L-functions associated with various mathematical objects (Dirichlet characters, elliptic curves, modular forms)
• L-functions provide a unifying framework for studying arithmetic properties and proving important conjectures (Birch and Swinnerton-Dyer conjecture, Langlands program)

Basic Properties of Zeta Functions

• Zeta functions are defined as infinite series or products indexed by natural numbers or prime numbers
• Riemann zeta function: $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} (1 - p^{-s})^{-1}$ for $\Re(s) > 1$
• Convergence of zeta functions depends on the complex variable $s$
  • Absolute convergence for $\Re(s) > 1$
  • Conditional convergence for $0 < \Re(s) \leq 1$
• Zeta functions have a pole at $s = 1$ with residue 1
• Satisfy functional equations relating values at $s$ and $1-s$
  • Example: $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
• Exhibit special values at integer arguments (Bernoulli numbers, values of the Riemann zeta function at even positive integers)
• Analytic continuation allows extending the domain of zeta functions to the entire complex plane (except for the pole at $s = 1$)

Riemann Zeta Function: Key Concepts

• The Riemann zeta function is the most well-known and extensively studied zeta function
• Defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\Re(s) > 1$
• Analytic continuation extends the definition to the entire complex plane (except for the pole at $s = 1$)
• Non-trivial zeros of the Riemann zeta function are complex numbers $\rho$ such that $\zeta(\rho) = 0$ and $0 < \Re(\rho) < 1$
• Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real part equal to $\frac{1}{2}$
  • Equivalent to the statement that the prime counting function $\pi(x)$ satisfies $\pi(x) = \text{Li}(x) + O(\sqrt{x} \log x)$
• Euler product formula: $\zeta(s) = \prod_{p \text{ prime}} (1 - p^{-s})^{-1}$ for $\Re(s) > 1$
  • Connects the Riemann zeta function to the distribution of prime numbers
• Special values: $\zeta(2) = \frac{\pi^2}{6}$, $\zeta(-1) = -\frac{1}{12}$ (Ramanujan summation)

L-Functions: Definition and Examples

• L-functions are generalizations of the Riemann zeta function associated with various mathematical objects
• Dirichlet L-functions: associated with Dirichlet characters $\chi$ modulo $q$
  • Defined as $L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}$ for $\Re(s) > 1$
  • Analytic continuation and functional equation similar to the Riemann zeta function
• Dedekind zeta functions: associated with number fields $K$
  • Defined as $\zeta_K(s) = \sum_{\mathfrak{a} \subset \mathcal{O}_K} \frac{1}{N(\mathfrak{a})^s}$ for $\Re(s) > 1$, where $\mathfrak{a}$ runs over non-zero ideals of the ring of integers $\mathcal{O}_K$ and $N(\mathfrak{a})$ is the norm of the ideal
• L-functions of elliptic curves: associated with elliptic curves $E$ over $\mathbb{Q}$
  • Defined as $L(E, s) = \prod_p L_p(E, p^{-s})^{-1}$, where $L_p(E, T)$ is the local factor at the prime $p$
  • Birch and Swinnerton-Dyer conjecture relates the rank of the elliptic curve to the order of vanishing of $L(E, s)$ at $s = 1$
• Artin L-functions: associated with representations of the absolute Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
• Automorphic L-functions: associated with automorphic representations of reductive groups

Analytic Continuation and Functional Equations

• Analytic continuation is the process of extending the domain of a complex function beyond its initial region of definition
• Zeta functions and L-functions are initially defined as convergent series for $\Re(s) > 1$
• Analytic continuation allows extending the definition to the entire complex plane (except for poles)
• Functional equations relate the values of zeta functions and L-functions at $s$ and $1-s$
  • Example for the Riemann zeta function: $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
• Functional equations are crucial for studying the behavior of zeta functions and L-functions in the critical strip $0 < \Re(s) < 1$
• Analytic continuation and functional equations are often proved using the Mellin transform and the Poisson summation formula
• The Riemann Hypothesis and its generalizations are statements about the location of zeros of zeta functions and L-functions in the critical strip

Connections to Number Theory

• Zeta functions and L-functions encode arithmetic information about various mathematical objects
• The Riemann zeta function is closely related to the distribution of prime numbers
  • Prime Number Theorem: $\pi(x) \sim \frac{x}{\log x}$ as $x \to \infty$, where $\pi(x)$ is the prime counting function
  • Equivalent to the statement that the Riemann zeta function has no zeros on the line $\Re(s) = 1$
• Dirichlet L-functions are used in the proof of Dirichlet's theorem on primes in arithmetic progressions
• L-functions of elliptic curves are related to the rank of the elliptic curve and the Birch and Swinnerton-Dyer conjecture
• Dedekind zeta functions provide information about the class number and unit group of number fields
• The Langlands program seeks to unify the study of zeta functions and L-functions associated with various mathematical objects (Galois representations, automorphic forms, algebraic varieties)
• Zeta functions and L-functions appear in the formulation of many important conjectures in number theory (Riemann Hypothesis, Generalized Riemann Hypothesis, Birch and Swinnerton-Dyer conjecture, Langlands functoriality)

Applications and Open Problems

• Zeta functions and L-functions have applications in various areas of mathematics and beyond
• Cryptography: the Riemann Hypothesis and its generalizations have implications for the security of cryptographic protocols based on prime numbers and elliptic curves
• Coding theory: zeta functions and L-functions appear in the study of error-correcting codes and their properties
• Physics: zeta functions and L-functions arise in quantum field theory (regularization of divergent series) and string theory (Riemann surfaces and modular forms)
• Open problems and conjectures related to zeta functions and L-functions:
  • Riemann Hypothesis: all non-trivial zeros of the Riemann zeta function have real part equal to $\frac{1}{2}$
  • Generalized Riemann Hypothesis: all non-trivial zeros of Dirichlet L-functions have real part equal to $\frac{1}{2}$
  • Birch and Swinnerton-Dyer conjecture: relates the rank of an elliptic curve to the order of vanishing of its L-function at $s = 1$
  • Langlands functoriality: conjectures about the relationships between L-functions associated with different mathematical objects
• Computational aspects: efficient algorithms for computing zeta functions and L-functions, numerical verification of conjectures