Dirichlet L-function

A complex function defined by a Dirichlet series whose coefficients are values of a Dirichlet character. These functions generalize the Riemann zeta function and encode arithmetic information about primes in residue classes.

Dirichlet character: A periodic, completely multiplicative arithmetic function that vanishes on integers not coprime to its modulus. Its values supply the coefficients in the defining series and determine each member of the family.
Dirichlet's theorem on arithmetic progressions: The theorem that every residue class coprime to a modulus contains infinitely many primes. Nonvanishing of Dirichlet L-functions at s=1 drives the theorem's proof.
Riemann zeta function: The function ζ(s)=sum n^(-s), initially defined for complex s with real part greater than one. The principal character recovers the zeta function up to factors from primes dividing the modulus.
Hurwitz zeta function: The function ζ(s,a)=sum (n+a)^(-s), which generalizes the Riemann zeta function by a shift parameter. It can express Dirichlet L-functions through residue-class sums, but is not itself character-defined.
Dirichlet series: A series of the form sum a(n)n^(-s), used to encode arithmetic sequences as complex functions. The defining formula is a Dirichlet series with character values as coefficients.
Prime number theorem for arithmetic progressions: An asymptotic formula describing how primes are distributed among reduced residue classes. Zeros of these functions control the error in counting primes across residue classes.
Complex analysis: The study of functions of complex variables and their analytic properties. Convergence, poles, zeros, and continuation are central to understanding these functions.
Hecke L-function: An L-function built from a Hecke character of a number field or its ideals. It extends the character-based construction from rational integers to broader number fields.
Euler product: A factorization of a Dirichlet series into a product indexed by prime numbers. Multiplicativity turns the series into local factors that expose prime-by-prime structure.
Class number formula: A formula relating the class number of a number field to special values of zeta or L-functions. Special values of quadratic Dirichlet L-functions encode class numbers of quadratic fields.

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