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 (∞ Xd |n(3 n/d)(8 n/d)d1) q n, ∞∑n=1 (∞d n (3 n/d)(8 d))q n, ∞(n=1 (∑d |(?3 d)(8 n/d)d)) q n, ∞∑ n=1 )(∑d | n)3 d(8)d)d q n, is determined in terms of Ramanujan's theta function.

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Keywords local densities, Quaternary quadratic forms, representations, Siegel's mass formula, theta functions
Persistent URL dx.doi.org/10.1142/S1793042112500972
Journal International Journal of Number Theory
Citation
Alaca, A, & Williams, K.S. (2012). 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. International Journal of Number Theory, 8(7), 1661–1686. doi:10.1142/S1793042112500972