2010/05/31 by Jeffrey C. Lagarias, Wen-Ching Winnie Li, W. -C. Winnie Li
Chemistry · Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics #Mathematics #Molecular spectroscopy and chirality #Physics #math.NT #msc:11M35
paper · pdf · doi:10.1515/form.2011.048
published as Forum Math. 24 (2012), no. 1, 49-84 · 29 pages, 3 figures; v2 notation changes, homotopy action on left
arxiv created 2010/06/23 · openalex publication_date 2010/07/08 · arxiv updated 2015/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract. This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. The Lerch zeta function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mrow> <m:mo>(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mo>=</m:mo> <m:msubsup> <m:mo/> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi/> </m:msubsup> <m:mfrac> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mi/> <m:mi>i</m:mi> <m:mi>n</m:mi> <m:mi>a</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mi>c</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> </m:mfrac> </m:mrow> </m:math> ζ (s,a,c) := ∑ n=0^∞ \frace2 π in a (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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mo>(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> 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:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mo>:</m:mo> <m:mo>=</m:mo> <m:mo/> <m:mo>(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo>)</m:mo> <m:mo/> <m:mi/> <m:mo/> <m:mo>(</m:mo> <m:mi/> <m:mo/> <m:mi/> <m:mo>)</m:mo> <m:mo/> <m:mo>(</m:mo> <m:mi/> <m:mo/> <m:mi/> <m:mo>)</m:mo> <m:mo/> <m:mo>,</m:mo> </m:mrow> </m:math> \mathcal M:= \lbrace (s, a, c) ∈ \mathbb C× ( \mathbb C∖ \mathbb Z) × ( \mathbb C∖ \mathbb Z) \rbrace , and that this analytic continuation becomes single-valued on the maximal abelian cover of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi/> </m:math> \mathcal M . We compute the monodromy functions describing the multivalued nature of this function on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi/> </m:math> \mathcal M , and determine various of its properties.