The Lerch Zeta function III. Polylogarithms and special values

Abstract

This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent (s, z, c) := ∞ n=0 z n (n+c) s is obtained from the Lerch zeta function ζ (s, a, c) by the change of variable z = e 2πia . 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 × (P 1 (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 Li m (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 Li m (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 Li m (z).

On certain zeta functions associated with Beatty sequences

“…It also bears the name of Lerch [19], who showed that for ℑz > 0 and q ∈ (0, 1) the functional equation ζ(z, q; 1 − s) = (2π) −s Γ(s) e( 1 4 s − zq)ζ(−q, z; s) + e(− 1 4 s + zq)ζ(q, 1 − z; s) holds; this is called Lerch's transformation formula. For an interesting account of the analytic properties of the Lipschitz-Lerch zeta function and related functions, we refer the reader to the work of Lagarius and Li [15][16][17][18]; see also Apostol [2].…”

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

“…In both cases, the meromorphic extension is continuous on its domain of holomorphy as a function in the two variables (s, w). We refer to [26] for much deeper knowledge on the meromorphic continuability of the Lerch transcendent as a function of all three variables. It suffices to show that for a, b ∈ A and p ∈ P a,b the map Recall that (without loss of generality) E a and E b are subsets of C. For M ∈ N 0 , let…”

The Lerch zeta function IV. Hecke operators

“…This property was noted in Parts II and III ( [29,30]). When restricting to real variables, we regard ∂ ∂a and ∂ ∂c as real differential operators.…”

“…The Hurwitz zeta function inherits the discontinuities of the Lerch zeta function at integer values of c. Milnor [32, p. 281] noted that at the value s = 1 the space K s includes on (0, 1) the odd function c − [29,30]) for which the Lerch zeta function is an eigenfunction.…”

“…When restricting to real variables, we regard ∂ ∂a and ∂ ∂c as real differential operators. (They were treated as complex differential operators in [29] and [30].) (2) In particular, it satisfies two four-term functional equations encoding a discrete symmetry under (s, a, c) to (1 − s, 1 − c, a).…”