A Dirichlet L-series is a series of the form L_k(s, χ) congruent sum_(n = 1)^∞ χ_k(n) n^(-s), where the number theoretic character χ_k(n) is an integer function with period k, are called Dirichlet L-series. These series are very important in additive number theory (they were used, for instance, to prove Dirichlet's theorem), and have a close connection with modular forms. Dirichlet L-series can be written as sums of Lerch transcendents with z a power of e^(2π i/k). Dirichlet L-series is implemented in the Wolfram Language as DirichletL[k, j, s] for the Dirichlet character χ(n) with modulus k and index j.
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