Sums of the triple divisor function over values of a ternary quadratic form

Ramanujan’s Ternary Quadratic Form

do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from representing N. By the classical theory of quadratic forms, it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer. Consequently if a genus consi...

Kaplansky’s Ternary Quadratic Form

This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, thenN is square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L-function of a suitable elliptic curve, a...

A characterization of triple semigroup of ternary functions and Demorgan triple semigroup of ternary functions

Gauss Sums & Representation by Ternary Quadratic Forms

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...

On the Exponent of Distribution of the Ternary Divisor Function

For any positive integer k ě 1, we denote by dk the k–fold divisor function: for n a positive integer, dkpnq is the number of solutions of the equation n “ n1 . . . nk, where the ni are positive integers. The purpose of this paper is to investigate the exponent of distribution of the ternary divisor function d3 in arithmetic progressions. More generally, we will say that a real number Θ ą 0 is ...