Dirichlet's theorem on primes in arithmetic progressions is a cornerstone of analytic number theory. It proves there are infinitely many primes in sequences like 3, 7, 11, 15, ... where each term differs by a fixed amount.

The proof hinges on Dirichlet characters and L-functions, which generalize ideas from the Riemann zeta function. By showing L(1,χ) ≠ 0 for non-principal characters, Dirichlet established his groundbreaking result on prime distributions.

Dirichlet Characters and L-functions

Fundamental Concepts of Dirichlet Characters

• Dirichlet characters define completely multiplicative functions from integers to complex numbers
• Characters modulo q map integers with the same residue modulo q to the same value
• Principal character χ₀ equals 1 for integers coprime to q and 0 otherwise
• Non-principal characters have the property that the sum of their values over a complete set of residues modulo q equals 0
• Orthogonality relations for Dirichlet characters play a crucial role in many proofs

L-functions and Their Properties

• L-functions associated with Dirichlet characters generalize the Riemann zeta function
• For a Dirichlet character χ, the L-function L(s, χ) defined as the sum of χ(n)/n^s for positive integers n
• L-functions converge absolutely for Re(s) > 1
• Euler product representation expresses L(s, χ) as a product over primes: L(s, χ) = ∏ᵨ (1 - χ(p)/p^s)^(-1)
• Analytic continuation extends L(s, χ) to the entire complex plane for non-principal characters
• For the principal character, L(s, χ₀) has a simple pole at s = 1, similar to the Riemann zeta function

Dirichlet's Theorem and Non-vanishing of L(1, χ)

Dirichlet's Theorem on Primes in Arithmetic Progressions

• States that for coprime positive integers a and q, there exist infinitely many primes in the arithmetic progression a + nq
• Proof relies on showing that L(1, χ) ≠ 0 for all non-principal characters modulo q
• Utilizes the logarithmic derivative of L(s, χ) to relate primes in arithmetic progressions to L-functions
• Demonstrates that the density of primes in each reduced residue class modulo q approaches 1/φ(q) as x approaches infinity

Non-vanishing of L(1, χ) and Its Implications

• L(1, χ) ≠ 0 for all non-principal characters χ modulo q forms a crucial step in proving Dirichlet's theorem
• Proof involves showing that the product of L(1, χ) over all characters χ modulo q is non-zero
• Utilizes properties of Gauss sums and the functional equation for L-functions
• Non-vanishing of L(1, χ) has implications beyond Dirichlet's theorem, including in the study of class numbers of number fields

Functional Equation for L-functions

• Relates values of L(s, χ) in the left half-plane to those in the right half-plane
• For primitive characters χ modulo q, the functional equation involves Gauss sums and gamma functions
• Takes the form L(1-s, χ̄) = W(χ)(q/π)^(s-1/2) Γ(s/2) L(s, χ) / Γ((1-s)/2), where W(χ) denotes a certain complex number of absolute value 1
• Allows for the study of L-functions in regions where the original series definition does not converge
• Plays a crucial role in the analytic continuation of L-functions and in proving the non-vanishing of L(1, χ)

Generalized Riemann Hypothesis

Statement and Implications of the Generalized Riemann Hypothesis

• Conjectures that all non-trivial zeros of L(s, χ) lie on the critical line Re(s) = 1/2 for all Dirichlet characters χ
• Generalizes the classical Riemann Hypothesis from the Riemann zeta function to all Dirichlet L-functions
• Would imply strong results about the distribution of primes in arithmetic progressions
• Provides upper bounds for the least prime in an arithmetic progression a + nq (Linnik's constant)
• Impacts various areas of number theory, including estimates for character sums and the distribution of quadratic residues

Connections to Other Mathematical Problems

• Relates to the Extended Riemann Hypothesis for zeta functions of number fields
• Influences the study of elliptic curves and their L-functions
• Connects to problems in computational number theory, such as primality testing and factorization algorithms
• Impacts the understanding of the distribution of Gaussian primes in the complex plane
• Provides insights into the behavior of multiplicative functions in number theory