TY - JOUR

T1 - The Lerch Zeta function III. Polylogarithms and special values

AU - Lagarias, Jeffrey C.

AU - Li, Wen Ching Winnie

N1 - Funding Information:
The authors thank Dinakar Ramakrishnan for conversations regarding his work on polylogarithms. The first author thanks Peter Scott for discussions and queries on multidimensional covering manifolds. The authors thank the reviewers for helpful comments. This project was initiated while the first author was at AT&T Labs-Research and the second author consulted there; they thank AT&T for support. The first author received support from the Mathematics Research Center at Stanford University in 2009–2010. The second author received support from the National Center for Theoretical Sciences and National Tsing Hua University in Taiwan in 2009–2014. To these institutions the authors express their gratitude. The research of the first author was supported by NSF grants DMS-1101373 and DMS-1401224 and that of the second author by NSF-grant DMS-1101368 and Simons Foundation grant No. 355798.
Publisher Copyright:
© 2016, The Author(s).

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85043268413&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85043268413&partnerID=8YFLogxK

U2 - 10.1186/s40687-015-0049-2

DO - 10.1186/s40687-015-0049-2

M3 - Article

AN - SCOPUS:85043268413

SN - 2522-0144

VL - 3

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

IS - 1

M1 - 2

ER -