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

Zeros of Systems of p{\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 $(2r)^r$.Comment: Revised version, with better treatment and results for characteristic

The largest prime factor of X3+2X^3+2

The largest prime factor of $X^3+2$ has been investigated by Hooley, who gave a conditional proof that it is infinitely often at least as large as $X^{1+\delta}$, with a certain positive constant $\delta$. It is trivial to obtain such a result with $\delta=0$. One may think of Hooley's result as an approximation to the conjecture that $X^3+2$ is infinitely often prime. The condition required by Hooley, his R$^{*}$ conjecture, gives a non-trivial bound for short Ramanujan-Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of $X^3+2$ is infinitely often at least as large as $X^{1+\delta}$, though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan-Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus

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