A Generalization of Epstein’s Zeta Function

Koecher defined in [3] the following zeta function associated with the matrix S(n) of a positive quadratic form and one complex variable ρ(1)

On the minimum of a positive quadratic form in n ( ≤ 8) variables (verification of Blichfeldt's calculations)

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated aseach inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.

The Construction of Perfect and Extreme Forms

Letbe a positive quadratic form of determinantD,and letMbe the minimum off(x) for integral x ≠ 0. Thenf(x) assumes the valueMfor a finite number of integral vectors x =±mk(k= 1 , … ,s)called theminimal vectors.

On perfect and extreme forms

Let(х) =(x1,x2, …xn) = ΣiΣiaij,xixf(aij= aij) be a positive quadratic form with determinantD, and letMbe the minimum offor integral x ≠ 0. Thenattains the valueMfor a finite number of integral x = ±mk( k = 1, …, s) called itsminimal vectors.