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Calculation and Applications of Epstein Zeta Functions

1975; American Mathematical Society; Volume: 29; Issue: 129 Linguagem: Inglês

10.2307/2005480

1088-6842

Daniel Shanks,

Advanced Algebra and Geometry

Rapidly convergent series are given for computing Epstein zeta functions at integer arguments.From these one may rapidly and accurately compute Dirichlet L functions and Dedekind zeta functions for quadratic and cubic fields of any negative discriminant.Tables of such functions computed in this way are described and numerous applications are given, including the evaluation of very slowly convergent products such as those that give constants of Landau and of Hardy-Littlewood. 1. Introduction.Many constants, such as those of Hardy-Littlewood [1], [2] and Landau [3], are given by very slowly convergent infinite products that can be transformed into rapidly convergent products containing the Dirichlet functions L{n, x) or La{n) for integer arguments n.Three-quarters of the latter can be obtained in closed form [4], but the computations become lengthy if a or n is large.The remaining, nonclosed-form L{n, x), such as f(3) or Catalan's constant L{2), can be computed by polylogarithms [5], polygamma functions, or other means [6] based upon the periodicity of the coefficients.But the period increases with the discriminant and, again, lengthy computations may be needed if accurate values are wanted.As a result, even constants of special interest such as «163 for the number of primes of the form «2 + « + 41, or bl4 for the number of numbers of the form u2 + 14u2, cf.[3, Eq. (5) and Section 4], have not been computed accurately prior to the present work.Values of such special constants are included below.If the algebraic field involved is nonabelian, such as 0) K = Ô(2!/3) or Q{31'3), the L{n, x) do not suffice, and one needs instead the Dedekind zeta functions $An) at integer arguments.It was the Bateman constants for (1) [7, especially Section 6] that led to the present investigation.Only now (eight years later) is this being published.For all cubic or quadratic fields K of negative discriminant -D, the wanted ^(s) or L{s, x) can be expressed in terms of Epstein zeta functions: