The Dirichlet beta function is defined by the sum β(x) | congruent | sum_(n = 0)^∞ (-1)^n (2n + 1)^(-x) | = | 2^(-x) Φ(-1, x, 1/2), where Φ(z, s, a) is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function ζ(x, a) by β(x) = 1/4^x[ζ(x, 1/4) - ζ(x, 3/4)].
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