A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.
In 1997 Jagy, Kaplansky and Schiemann determined that there are at most 913 (classes of) primitive, positive-definite, integral ternary quadratic forms ax2 + by2 + cz2 + dyz + ezx + fxy which are regular. In this paper the positive integers represented by these 913 ternary forms are given.
A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.
Abstract A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals.
We present an explicit evaluation of the double Gauss sum G(a,b,c;S;p^n):=∑_x,y=0^p^n-1 e^2π i S(ax^2+bxy+cy^2)/p^n, where a, b, c are integers such that (a,b,c)=1, p is a prime, n is a positive integer, and S is an integer coprime to p.
We express products of Lambert series as power series. We use these Lambert series-to-power series identities to obtain new Liouville identities with two functions. Many of the known Liouville identities follow from our new identities. We explore the relationships between Lambert series, new Liouville identities, and convolution sums. We then obtain recursive formulae for various convolution sums.
