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Register a new userOn representation of integers by binary quadratic forms
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Given a negative $D>-(log X)^{log 2-delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of discriminant $D$. We also give an analogous upper bound for square free integers of the form $q+a<X$ where $q$ is prime and $ainmathbb Z$ is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers $q+a<X$ represented by a binary quadratic form of discriminant $D$, where $D$ is allowed to grow with $X$ as above. An immediate consequence of this, coming from recent work of the authors in [BF], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.
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In this article, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given by Henryk Iwaniec and Ritabrata Munshi in cite{H-R}.
In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to be $C_0^infty$-smooth and to vanish near the singularity of the quadric. In out work we allow the weight function to be finitely smooth and not vanish near the singularity, and we give also an explicit dependence on the weight function.
It is shown that a system of $r$ quadratic forms over a ${mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$.
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the literature. Motivated by the works of Liu, we first establish many parameterized identities with two parameters by employing different $q$-transformation formulas and then deduce various Hecke-type identities associated with definite quadratic forms by specializing the choice of these two parameters. As applications, we utilize some of these Hecke-type identities to establish families of inequalities for several partition functions. Our proofs heavily rely on some formulas from the work of Zhi-Guo Liu.
In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
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