The number of representations of positive integers by positive quadratic forms

Abstract

In this paper, we consider the number of representations of pos- itive integers by direct sum of binary quadratic forms F1(x1; y1) = x2 1 + 3x1x2 + 8x2 2 and G1(x1; y1) = 2x2 1 + 3x1x2 + 4x2 2 of discriminant ¡23. We derived some results concerning the modular forms }(¿; Q; 'rs) and their or- ders ord (}(¿; Q; 'rs); i1; ¡0(23)), where Q (Q = F2;G2; F1 ©G1; direct sum of F1 and G1) is the positive de¯nite quadratic form and 'rs is the spherical function of second order with respect to Q. We constructed a basis for the cusp form space S4(¡0(23); 1), and then we obtained some formulas for the number of representations of positive integer n by positive de¯nite quadratic forms F4;G4; F3 © G1; F2 © G2 and F1 © G3 using the elements of the space S4(¡0(23); 1).