Lseries

L-series, or Dirichlet L-series, are complex functions attached to Dirichlet characters χ modulo q, defined by the Dirichlet series L(s, χ) = sum_{n=1}^∞ χ(n)/n^s for Re(s) > 1. They form multiplicative Dirichlet series with Euler products L(s, χ) = ∏_p (1 - χ(p) p^{-s})^{-1}, valid in the same region. The Riemann zeta function ζ(s) is the special case L(s, χ0) where χ0 is the principal character modulo 1; more generally principal characters modulo q yield L(s, χ0) = ζ(s) ∏_{p|q} (1 - p^{-s}), which has a simple pole at s = 1 with residue φ(q)/q.

For non-principal χ, L(s, χ) is entire. Primitive characters satisfy a functional equation relating L(s, χ) to L(1 − s, χ̄), involving gamma factors and Gauss sums. These analytic properties underpin many number-theoretic results and conjectures. Dirichlet L-series encode information about primes in arithmetic progressions: Dirichlet’s theorem asserts that primes are equidistributed among residue classes coprime to q, a fact reflected in the nontrivial zeros and special values of L(s, χ).

Beyond Dirichlet characters, L-series generalize to broader contexts in number theory, including Hasse–Weil L-series attached to algebraic varieties over number fields and automorphic L-functions central to the Langlands program. Special values of L(s, χ) at integers often relate to arithmetic data such as class numbers and regulators, highlighting the deep connections between analysis and arithmetic found in L-functions.