Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series

In this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Gamma be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Gamma\H has finite volume yet is non-compact. Associated to each cusp of Gamma\H, there is a classically studied first-order non-holomorphic Eisenstein series E(s,z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein series E(s,z) admits a meromorphic continuation with a simple pole at s =: 1. Classically, Kronecker's limit formula is the study of the constant term K-1(z) in the Laurent expansion of E(s,z) at s = 1. A number of authors recently have studied what is known as the second-order Eisenstein series E*(s, z), which is formed by twisting the Dirichlet series that defines the series E(s, z) by periods of a given cusp form f. In the work we present here, we study an analogue of Kronecker's limit formula in the setting of the second-order Eisenstein series E*(s,z), meaning we determine the constant term K-2(Z) in the Laurent expansion of E*(s, z) at its first pole, which is also at s = 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function K-1. We then define two families of convolution Dirichlet series, denoted by L-m(+) and L-m(-) with m epsilon N, which are formed by using the Fourier coefficients of K-1 and the weight two cusp form f. We prove that for all m, L-m(+) and L-m(-) admit a meromorphic continuation and are holomorphic at s = 1. Turning our attention to the second-order Kronecker limit function K-2, we first express K-2 as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp form f, the Fourier coefficients of the first-order Kronecker limit function K-1, and special values L-m(+)(1) and L-m(-)(1) of the convolution Dirichlet series. Finally, we prove a bound for the special values L-m(+)(1) and L-m(-)(1) which then implies a bound for the Fourier coefficients Of K-2. 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.