The Lerch zeta function II. Analytic continuation

This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. The Lerch zeta function ζ(s, a, c): = Σ ^∞ _n=0 e ^2πina/ (n+c) ^s was introduced by Lipschitz in 1857, and is named after Lerch, who showed in 1887 that it satisfied a functional equation. Here we analytically continue ζ(s, a, c) as a function of three complex variables. We show that it is well-defined as a multivalued function on the manifold M : = {(s, a, c) ∈ ℂ × (ℂ\ℤ) × (ℂ \ ℤ)}, and that this analytic continuation becomes single-valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of its properties.