Generalized Zeta Functions

How does generalizing the Riemann zeta function to Dirichlet characters and Dedekind zeta functions allow a precise understanding of prime distribution in arithmetic progressions and the arithmetic structure of algebraic extensions of Q?

• **Cryptography:** Dirichlet's theorem on primes in progressions is used in RSA key generation (one may need a prime p with p = 3 mod 4 for certain schemes).
• **Langlands program:** Sarnak and collaborators (2022) proved new cases of Langlands correspondence. Over 100 leading mathematicians work on this unification of number theory, geometry, and physics.
• **Factoring algorithms:** Generalized GRH for L(s,chi) bounds the complexity of the General Number Field Sieve (GNFS) - the best known factoring algorithm.
• **Quantum field theory:** Geometric Langlands connects L-functions to D-modules and local systems, related to conformal blocks in WZW models.

Dirichlet L-Functions

Dirichlet L-functions generalize the Riemann zeta function by introducing a character chi - a multiplicative homomorphism from (Z/mZ)* to C*. Dirichlet proved in 1837 through analytic properties of L(s,chi) at s=1 that every arithmetic progression {a, a+m, a+2m, ...} with gcd(a,m)=1 contains infinitely many primes. The key step: L(1,chi) is nonzero for every non-trivial primitive character chi.

Special values: L(1, chi_{-4}) = 1 - 1/3 + 1/5 - ... = pi/4 (Leibniz formula). L(2, chi_{-4}) = G (the Catalan constant). L(3, chi_{-4}) = pi^3/32. For a quadratic character chi_D: L(1, chi_D) is expressed via logarithms of units in Q(sqrt(D)) - Dirichlet's class number formula. These special values are connected to periods of algebraic varieties through motives.

Generalized GRH for L(s,chi): all zeros in the critical strip lie on Re(s) = 1/2. Consequence: the least prime p with p equal to 1 mod q is at most O(q^2 log^2 q) under GRH (unconditionally only O(q^{5.5+eps})).

The Kronecker symbol (D/n) generalizes the Legendre symbol to all integers n. For D = -4: (-4/n) = 0 if n is even, 1 if n = 1 mod 4, -1 if n = 3 mod 4. This is a primitive character mod 4. The principal character chi_0 has L(1, chi_0) = infinity (pole via zeta(s)). Only non-trivial primitive characters give L(1, chi) finite and nonzero.

Dedekind Zeta and Number Fields

In 2022, Peter Sarnak and collaborators proved new cases of the Langlands correspondence, linking automorphic L-functions to Galois L-functions. The Dedekind zeta function zeta_K(s) is the analogue of the Riemann zeta for an arbitrary number field K, summing over non-zero ideals of the ring of integers O_K. The Langlands program in practice starts with GL(1) characters (Hecke Grossencharacters), then GL(2) (modular forms), and so on.

The Langlands program in practice: GL(1) - Hecke Grossencharacters; GL(2) - modular forms and 2-dimensional Galois representations (Deligne). The modularity theorem (Wiles, 1995) proves every elliptic curve over Q corresponds to a GL(2) automorphic form. Current frontiers: Sato-Tate conjecture (proved 2011), functoriality between GL groups, global Langlands for GL(n).

Langlands program and connections to physics

L-functions are the central objects of the Langlands program, which proposes a vast web of correspondences unifying number theory, geometry, and physics.

• Number theory — Related topic
• Representation theory — Related topic
• Algebraic geometry — Related topic
• Conformal field theory — Related topic

Итоги

• L(s, chi) = sum chi(n)/n^s - generalizes zeta(s) to Dirichlet characters; for primitive chi: L(1,chi) is nonzero, implying Dirichlet's theorem on primes in arithmetic progressions
• Functional equation for Lambda(s,chi): Lambda(s,chi) = epsilon(chi) Lambda(1-s, chi_bar); the root number epsilon is determined by the Gauss sum tau(chi) = sum chi(n) e^{2 pi i n/m}
• Dedekind zeta zeta_K(s) = prod (1-N(p)^{-s})^{-1} generalizes zeta(s) to number fields K via ideal norms
• For abelian extensions K/Q (Kronecker-Weber): zeta_K(s) = zeta(s) times product L(s,chi) - connects to Dirichlet L-functions
• Class number formula: residue of zeta_K at s=1 encodes h_K (class number), R_K (regulator), d_K (discriminant), and the number of roots of unity w_K