Bounds for the Cubic Weyl Sum
It is shown, subject to the abc-conjecture, that \[\sum_{n\le N}\exp(2πiαn^3)\ll_{ε,α}N^{5/7+ε}\] for any $ε>0$ and any quadratic irrational $α$.
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arXiv subjects
D. R. Heath-Brown
Publications and source records attributed to D. R. Heath-Brown.
Zeros of Systems of ${\mathfrak p}$-adic Quadratic Forms
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 $(2r)^r$.
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Pair Correlation for Fractional Parts of $αn^2$
We construct real numbers $α$ for which the pair correlation function \[N^{-1}#\{m<n\le N:||αm^2-αn^2||\le XN^{-1}\}\] tends to $X$ as $N$ grows. Moreover we show for any "Diophantine" $α$ that the pair correlation function is $X+O(X^{7/8})+O((\log N)^{-1}$ for $1\le X\le\log N$.
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A note on the fourth moment of Dirichlet L-functions
We prove an asymptotic formula for the fourth power mean of Dirichlet L-functions averaged over primitive characters to modulus q and over t\in [0,T] which is particularly effective when q \ge T. In this range the correct order of magnitude was not previously known.
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Convexity bounds for L-functions
We give a sharp convexity estimate for L-functions which have a functional equation and an Euler product.
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Sums and Differences of Three k-th Powers
Let k>2 be a fixed integer exponent and let θ> 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^θN^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take θstrictly less than 1. We also prove the estimate O(B^{10/k}), (subject to N << B) which is better for large k. The results extend to representations by an arbitrary fixed nonsingular ternary from. However ``non-trivial'' must then be suitably defined. Consideration of the singular form x^{k-1}y-z^k allows us to establish an asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes, answering a problem raised by Hooley.
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Rational points on quartic hypersurfaces
Let X be a non-singular projective hypersurface of degree 4, which is defined over the rational numbers. Assume that X has dimension 39 or more, and that X contains a real point and p-adic points for every prime p. Then X is shown to contain infinitely many rational points.
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Density of non-residues in Burgess-type intervals and applications
We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.
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Integral points on cubic hypersurfaces
Let g be a cubic polynomial with integer coefficients and n>9 variables, and assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k. We show that the equation g=0 has infinitely many integer solutions when the cubic part of g defines a projective hypersurface with singular locus of dimension <n-10. The proof is based on the Hardy-Littlewood circle method.
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The density of rational points on non-singular hypersurfaces, I
Let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $d>1$, and let $ε>0$. This paper is concerned with the conjecture that there are $O(B^{n-1+ε})$ rational points on $X$ that have height at most $B$, in which the implied constant is allowed to depend only upon $d, ε$ and $n$. In particular this conjecture is shown to hold as soon as $d>4$. Furthermore, the main ideas in the proof are used to obtain new paucity estimates for certain diophantine equations.
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The density of rational points on non-singular hypersurfaces, II
This paper establishes the conjecture that a non-singular projective hypersurface of dimension $r$, which is not equal to a linear space, contains $O(B^{r+ε})$ rational points of height at most $B$, for any choice of $ε>0$. The implied constant in this estimate depends at most upon $ε, r$ and the degree of the hypersurface.
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Simultaneous equal sums of three powers
Using a recent result of Salberger, we establish the paucity of non-trivial positive integer solutions to a certain system of diagonal Diophantine equations.
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Plane curves in boxes and equal sums of two powers
Given an absolutely irreducible ternary form $F$, the purpose of this paper is to produce better upper bounds for the number of integer solutions to the equation F=0, that are restricted to lie in very lopsided boxes. As an application of the main result, a new paucity estimate is obtained for equal sums of two like powers.
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Counting rational points on algebraic varieties
Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and fixed dimension, and are essentially best possible for varieties of degree at least six.
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Counting rational points on hypersurfaces
Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever $n<6$, or whenever the hypersurface is not a union of lines, we obtain estimates that are essentially best possible and that are uniform in $d$ and $n$.
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The density of rational points in curves and surfaces
Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes from the counting function those points that lie on lines in the surface. The bounds are uniform for all $X$ of a given degree. They are best possible in the case of curves. As an application it is shown that if $F$ is an irreducible binary form of degree 3 or more then almost all integers represented by $F$ have essentially one such representation.
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The average analytic rank of elliptic curves
All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over Q is at most 2, thereby improving a result of Brumer. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld. A third result concerns the density of curves with analytic rank at least R, and shows that the proportion of such curves decreases faster than exponentially as R grows. The proofs depend on an analogue of Weil's ``explicit formula''.
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The density of rational points on Cayley's cubic surface
The Cayley cubic surface is given by the equation sum_{i=1}^4 X_i^{-1}=0. We show that the number of non-trivial primitive integer points of size at most B is of exact order B(log B)^6, as predicted by Manin's conjecture.
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