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

1. Many important applications of analysis to number-theory require the study of a functionf (s) of a complex variables = σ +i τ near a singular points 0 = σ0 +i τ0 . The functionsf (s) is frequently defined for σ > σ0 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 powers . 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 appropriatef (s) at a simple poles 0 . A deeper question then suggested is that of finding lims →s 0 (f (s) — R/s-s 0 ). In particular, Kronecker's fundamental formula arises whenf (s) is a homogeneous binary quadratic form in the variables of summation. Thus, let aa (≠ 0),b, c be any constants real or complex which are such that the roots ω1 , ω2 of the quadratic formϕ (x, y) =ax 2 +bxy +cy 2 =a (x - ω1 y ) (x - ω2 y ) are neither real nor equal. We need only distinguish the two cases (I) I (ω1 ) > 0, I (ω2 ) < 0, (II) I (ω1 ) > 0, I (ω2 ) > 0, as the others can be included by writing —y fory .