Related papers: Binary linear forms as sums of two squares

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the…

The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Ch\^atelet surfaces defined over the rationals.

We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…

We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over the rationals.

The problem of simultaneous decomposition of binary forms as sums of powers of linear forms is studied. For generic forms the minimal number of linear forms needed is found and the space parametrizing all the possible decompositions is…

Let $n$ be a positive integer. In this paper we estimate the size of the set of linear forms $b_1\log a_1 + b_2\log a_2+...+b_n\log a_n$, where $|b_i|\leq B_i$ and $1\leq a_i\leq A_i$ are integers, as $A_i,B_i\to \infty$.

Let $s_1, s_2, \ldots$ be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant $D$. We show that \[ \limsup_{n \rightarrow \infty} \frac{s_{n+1}-s_n}{\log…

Given an integer $d \ge 2$, what is the least $r$ so that there is a set of binary quadratic forms $\{f_1,\dots,f_r\}$ for which $\{f_j^d\}$ is non-trivially linearly dependent? We show that if $r \le 4$, then $d \le 5$, and for $d \ge 4$,…

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions $a, b \in \mathbb N$ to the equations $a+b=n$ and $a-b=n$, where $a$ is $k$-free and $b$ is $l$-free. This…

We continue our recent work on additive problems with prime summands: we already studied the \emph{average} number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with…

H. J. S. Smith proved Fermat's two-square theorem using the notion of palindromic continuants. In this paper we extend Smith's approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of…

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…

Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb…

We generalise the square sieve developed by Heath-Brown to higher powers in order to improve on the error term for the problem of counting consecutive power-free numbers.

We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of…

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms,…

We analyze the average behavior of various arithmetic functions at the values of degree $d$ binary forms ordered by height, with probability $1$. This approach yields averaged versions of the Chowla conjecture and the Bateman-Horn…