The minima of quadratic forms and the behavior of Epstein and Dedekind zeta functions

Hecke's correspondence between modular forms and Dirichlet series is put into a quantitative form giving expansions of the Dirichlet series in series of incomplete gamma functions in two special cases. The expansion is applied to show, for example, the positivity of Epstein's zeta function at $\text{s =}\text{n}\text{4}$ when the n-ary positive real quadratic form involved has a “small” minimum over the integer lattice. Hecke's integral formula is used to consider consequences for the Dedekind zeta function of a number field.