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 ${\mathrm{\Gamma }}_0(N)$ as regularized theta lifts of a single type of Poincaré series, due to Selberg. Using this realization and properties of the Poincaré 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 Poincaré series.

经典的Kronecker极限公式描述了与模群的无穷大相关的非全同型爱森斯坦级数的一阶极点上的Laurent展开中的常数项。最近，Jorgenson，Kramer和第一作者的研究针对与第一类Fuchsian群的双曲和椭圆形元素有关的非全同型Eisenstein级数研究了亚纯连续性和Kronecker极限类型公式。在当前的工作中，我们实现了所有三种类型的爱森斯坦级数的平均版本，用于${\mathrm{\Gamma }}_0（ñ）$由于塞尔伯格的缘故，它是单一类型庞加莱系列的正则theta升降机。利用庞加莱级数的这种认识和性质，我们推导了上述爱森斯坦级数的亚纯连续性和Kronecker极限公式。然后，通过与基础庞加莱级数的特殊值相关的Borcherds乘积的绝对值的对数，给出相应的Kronecker极限函数。