Convolution Dirichlet Series and a Kronecker Limit Formula for Second-Order Eisenstein Series

Abstract

AbstractIn this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Γ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Γ\H has finite volume yet is non-compact. Associated to each cusp of Γ\H, there is a classically studiedfirst-ordernon-holomorphic Eisenstein seriesE(s, z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein seriesE(s, z) admits a meromorphic continuation with a simple pole ats= 1. Classically, Kronecker’s limit formula is the study of the constant term1(z) in the Laurent expansion ofE(s, z) ats= 1. A number of authors recently have studied what is known as thesecond-orderEisenstein seriesE*(s, z), which is formed by twisting the Dirichlet series that defines the seriesE(s, z) by periods of a given cusp formf. In the work we present here, we study an analogue of Kronecker’s limit formula in the setting of the second-order Eisenstein seriesE* (s, z), meaning we determine the constant term2(z) in the Laurent expansion ofE*(s, z) at its first pole, which is also ats= 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function1. We then define two families of convolution Dirichlet series, denoted byandwithm∈ ℕ, which are formed by using the Fourier coefficients of1and the weight two cusp formf. We prove that for allm,andadmit a meromorphic continuation and are holomorphic ats= 1. Turning our attention to the second-order Kronecker limit function2, we first express2as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp formf, the Fourier coefficients of the first-order Kronecker limit function1, and special values(1) and(1) of the convolution Dirichlet series. Finally, we prove a bound for the special values(1) and(1) which then implies a bound for the Fourier coefficients of2. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.