The Lerch Zeta function III. Polylogarithms and special values

Jeffrey C. Lagarias, Wen Ching Winnie Li

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Abstract

This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent Φ(s,z,c):=∑n=0∞zn(n+c)s is obtained from the Lerch zeta function ζ(s, a, c) by the change of variable z= e2 π i a. We show that it analytically continues to a maximal domain of holomorphy in three complex variables (s, z, c), as a multivalued function defined over the base manifold C× (P1(C) \ { 0 , 1 , ∞}) × (C\ Z) and compute the monodromy functions describing the multivaluedness. For positive integer values s= m and c= 1 this function is closely related to the classical m-th order polylogarithm Lim(z). We study its behavior as a function of two variables (z, c) for “special values” where s= m is an integer. For m≥ 1 we show that it is a one-parameter deformation of Lim(z) , which satisfies a linear ODE, depending on c∈ C, of order m+ 1 of Fuchsian type on the Riemann sphere. We determine the associated (m+ 1) -dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of Lim(z).

Original languageEnglish (US)
Article number2
JournalResearch in Mathematical Sciences
Volume3
Issue number1
DOIs
Publication statusPublished - Dec 1 2016

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Mathematics (miscellaneous)
  • Computational Mathematics
  • Applied Mathematics

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