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

Abstract

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 $\mathbf{H}$ such that the quotient $\Gamma \backslash \mathbf{H}$ has finite volume yet is non-compact. Associated to each cusp of $\Gamma \backslash \mathbf{H}$, there is a classically studied {\it 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 $\mathcal{K}_{1}(z)$ in the Laurent expansion of $E(s, z)$ at $s = 1$. A number of authors recently have studied what is known as the {\it second-order\/} Eisenstein series $E^{\ast}(s, z)$, 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 $E^{\ast}(s, z)$, meaning we determine the constant term $\mathcal{K}_{2}(z)$ in the Laurent expansion of $E^{\ast}(s, z)$ 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 $\mathcal{K}_{1}$. We then define two families of convolution Dirichlet series, denoted by $L^{+}_{m}$ and $L^{-}_{m}$ with $m \in \mathbb{N}$, which are formed by using the Fourier coefficients of $\mathcal{K}_{1}$ 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 $\mathcal{K}_{2}$, we first express $\mathcal{K}_{2}$ 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 $\mathcal{K}_{1}$, 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 $\mathcal{K}_{2}$. 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.