Related papers: Simultaneous Integer Values of Pairs of Quadratic …

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its…

A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all…

In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simultaneous inequalities we require at least $m+2$ variables,…

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach…

We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.

Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

A quadratic form f is said to have semigroup property if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all binary integer quadratic forms with semigroup property. If…

Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q=0, which lie in a box with sides of length 2B, as B tends to infinity. The estimates…

Let $K$ be a number field. The \textit{integral trace form} is the integral quadratic form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral…

We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously unsolved cases of quartic and quintic fields. This fully…

Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence…

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

Given $A\subseteq \mathbb{Z}$, the ratio set or the quotient set of $A$ is defined by $R(A):=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attained…

We prove that any function with real-valued coefficients, whose input is 4 binary variables and whose output is a real number, is perfectly equivalent to a quadratic function whose input is 5 binary variables and is minimized over the new…

Random coupled parabolic partial differential models are solved numerically using random cosine Fourier transform together with non Gaussian random numerical integration that capture the highly oscillatory behavior of the involved…

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number…