ON THE QUATERNARY FORMS x~2 + y~2 + 2z~2 + 3t~2,x~2 + 2y~2 + 2z~2 + 6t~2, x~2 + 3y~2 + 3z~2 + 6t~2 AND 2x~2 + 3y~2 + 6z~2 + 6t~2

摘要

Formulas are proved for the number of representations of a positive integer by each of the four quaternary quadratic forms x~2 +y~2+2z~2+3t~2, x~2+2y~2+2z~2+6t~2, x~2+3y~2+3z~2+6t~2 and 2x~2 + 3y~2 + 6z~2 + 6t~2. As a consequence of these formulas, each of the four series ∑_(n=1)~∞(∑_(d|n)((-3)/(n/d))((-8)/(n/d))d)q~n,∑_(n=1)~∞(∑_(d|n)((-3)/(n/d))((-8)/d)d)q~n,∑_(n=1)~∞(∑_(d)((-3)/d)((-8)/(n/d))d)q~n,∑_(n=1)~∞(∑_(d)((-3)/d)((-8)/d))q~n,is determined in terms of Ramanujan's theta function.