On the Number of Representations of Positive Integers by Some Quadratic Forms in Fourteen Variables

Abstract

A way of finding explicit exact formulas for the number of representations of positive integers by some diagonal positive quadratic forms in fourteen variables is suggested.

On the number of representations of positive integers by quadratic forms as the basis of the space S4(Γ0(47), 1)

“…are spherical functions of second order with respect to the positive definite quadratic form Q in k variables [1].…”

“…Let F 1 = ax 2 1 + bx 1 x 2 + cx 2 2 and G 1 = dx 2 1 + ex 1 x 2 + f x 2 2 be two positive definite quadratic forms with discriminant ∆(F 1 ) and ∆(G 1 ), respectively. For each k 1, let F k and G k denote the direct sum of k-copies of F 1 and G 1 , respectively, where F 1 and G 1 have two variables, F 2 and G 2 have four variables, and therefore F k and G k have 2k variables.…”