# Kronecker limit formulas for parabolic, hyperbolic, and elliptic Eisenstein series via Borcherds products

## TL;DR

This paper derives Kronecker limit formulas for parabolic, hyperbolic, and elliptic Eisenstein series on modular groups using Borcherds products, unifying their analysis through Poincaré series and theta lifts.

## Contribution

It introduces a unified approach to Kronecker limit formulas for all three types of Eisenstein series via regularized theta lifts and Borcherds products.

## Key findings

- Derived meromorphic continuation for Eisenstein series.
- Established Kronecker limit formulas for all three types.
- Expressed limit functions as Borcherds product logarithms.

## Abstract

The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for $\Gamma_0(N)$ as regularized theta lifts of a single type of Poincar\'e series, due to Selberg. Using this realization and properties of the Poincar\'e series we derive the meromorphic continuation and Kronecker limit formulas for the above Eisenstein series. The corresponding Kronecker limit functions are then given by the logarithm of the absolute value of the Borcherds product associated to a special value of the underlying Poincar\'e series.