Kronecker's fundamental limit formula in the theory of numbers and elliptic functions, and similar theorems

Abstract

1. Many important applications of analysis to number-theory require the study of a function f (s) of a complex variable s = σ + i τ near a singular point s 0 = σ 0 + i τ 0 . The functions f (s) is frequently defined for σ > σ 0 by an infinite series, really d Dirichlet's series, the general term of which is a function of the variables of summation, e. g ., a quadratic form, raised to the power s . Thus the question of finding the number of classes of binary quadratic forms of given determinant, or the number of classes of ideals in a given field, depends upon the residue, Say R, of an appropriate f (s) at a simple pole s 0 . A deeper question then suggested is that of finding lim s → s 0 ( f (s) — R/ s-s 0 ). In particular, Kronecker's fundamental formula arises when f (s) is a homogeneous binary quadratic form in the variables of summation. Thus, let a a (≠ 0), b, c be any constants real or complex which are such that the roots ω 1 , ω 2 of the quadratic form ϕ (x, y) = ax 2 + bxy + cy 2 = a ( x - ω 1 y ) ( x - ω 2 y ) are neither real nor equal. We need only distinguish the two cases (I) I (ω 1 ) > 0, I (ω 2 ) < 0, (II) I (ω 1 ) > 0, I (ω 2 ) > 0, as the others can be included by writing — y for y .