The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation(1)

$\begin{equation*}r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty).\end{equation*}$

${\mathfrak s}$

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.

We establish that almost all natural numbers n are the sum of four cubes of positive integers, one of which is no larger than n5/36 . The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.

Among the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.

We prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can be written as the sum of eight cubes of P2-numbers.

As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp forms of half-integral weight, it is shown that any sufficiently large number $n\equiv 3$ (mod 24) with $5 \nmid n$ is expressible as a sum of three squares of integers having at most 521 prime factors.

Consider a system of diagonal equations \begin{equation}\sum_{j=1}^sa_{ij}x_j^k=0\quad (1\le i\le r),\end{equation} satisfying the property that the (fixed) integral coefficient matrix $(a_{ij})$ contains no singular $r\times r$ submatrix. A recent paper of the authors [3] establishes that whenever $k\ge 3$ and $s>(3r+1)2^{k-2}$, then the expected asymptotic formula holds for the number $N(P)$ of integral solutions ${\bf x}$ of ($1{\cdot}1$) with $|x_i|\le P$$(1\le i\le s)$.

This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [−P,P]s, provided only that s > (3r+1)2k−2. By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r2k.

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection, these estimates are applied to derive anupper bound for the $s$th moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as $s\ge 7.691$. Related arguments demonstrate that all large integers $n$ are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most $\exp (c(\log n\log \log n)^{1/2})$, for a suitable positive number $c$. This conclusion improves a previous result of G. Harcos in whichnine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

Additive representations of natural numbers by mixtures of squares, cubes and biquadrates belong to the class of more interesting special cases which form the object of attention for testing the general expectation that any sufficiently large natural number n is representable in the form

formula here

as soon as the reciprocal sum [sum ]sj=1k−1j is reasonably large. With the exception of a handful of very special problems, in the current state of knowledge the latter reciprocal sum must exceed 2, at the very least, in order that it be feasible to successfully apply the Hardy–Littlewood method to treat the corresponding additive problem. Here we remove a case from the list of those combinations of exponents which have defied treatment thus far.

Let k ≥ 2 be an integer. Let Ek (N) be the number of natural numbers not exceeding N which are not the sum of a prime and a k-th power of a natural number. Assuming the Riemann Hypothesis for all Dirichlet L-functions it is shown that Ek (N) ≪ N1-1/25k.

A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shape

where (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.

It was shown by Davenport and Roth [7] that the values taken by

at integer points ( x1, …, x8) ∈ ℤ8 are dense on the real line, providing at least one of the ratios λi/λj, is irrational. Here and throughout, λi denote such nonzero real numbers. More precisely, Liu, Ng and Tsang [8] showed that for all the inequality

has infinitely many solutions in integers. Later Baker [1] obtained the same result in the enlarged range . In this note we improve this further, the progress being considerable.

The determination of the minimal s such that all large natural numbers n admit a representation as

is an interesting problem in the additive theory of numbers and has a considerable literature, For historical comments the reader is referred to the author's paper [2] where the best currently known result is proved. The purpose here is a further improvement.