ON THE QUATERNARY FORMS x²+y²+2z²+3t², x²+2y²+2z²+6t², x²+3y²+3z²+6t² AND 2x²+3y²+6z²+6t²

Abstract

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 [Formula: see text] is determined in terms of Ramanujan's theta function.