The density of zeros of forms for which weak approximation fails

The weak approximation principal fails for the forms x3 + y3 + z3 = kw3, when k = 2 or 3. The question therefore arises as to what asymptotic density one should predict for the rational zeros of these forms. Evidence, both numerical and theoretical, is presented, which suggests that, for forms of the above type, the product of the local densities still gives the correct global density. Let f(x1,..., xn) ∈ Q[x1,..., xn] be a rational form. We say that f satisfies the weak approximation principle if the following condition holds. (WA): Given an ε> 0 and a finite set S of places of Q, and zeros (xν1,..., x ν n) ∈ Qnν of the form f, we can find a rational zero (x1,..., xn) of f such that, |xi − xνi |ν < ε, for 1 ≤ i ≤ n and ν ∈ S. Alternatively, we may write X(K) for the points on the hypersurface f = 0 whose coordinates lie in the field K, and consider the produc

Artin's Conjecture on Zeros of pp-Adic Forms

This is an exposition of work on Artin's Conjecture on the zeros of $p$-adic forms. A variety of lines of attack are described, going back to 1945. However there is particular emphasis on recent developments concerning quartic forms on the one hand, and systems of quadratic forms on the other.Comment: Submitted for publication as part of ICM 201

Lattice points in the sphere

Our goal in this paper is to give a new estimate for the number of integer lattice points lying in a sphere of radius R centred at the origin. Thus we define S(R) = #{x ∈ ZZ3: ||x| | ≤ R}

A mean value estimate for real character sums

There are a number of well known estimates for averages of Dirichlet polynomi-als. For example one has ∫

The distribution and moments of the error term in the Dirichlet divisor problem

This paper will consider results about the distribution and moments of some of the well known error terms in analytic number theory. To focus attention we begin by considering the error term ∆(x) in the Dirichlet divisor problem, which is defined a