Results 1 
8 of
8
Integers, without large prime factors, in arithmetic progressions, II
"... : We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fe ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of Sunit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...
On Sums of Seven Cubes
 Math. Comp
, 1999
"... Abstract. We show that every integer between 1290741 and 3.375 × 10 12 is a sum of 5 nonnegative cubes, from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 nonnegative cubes. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. We show that every integer between 1290741 and 3.375 × 10 12 is a sum of 5 nonnegative cubes, from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 nonnegative cubes.
Waring’s Problem: A Survey
"... tribus, &c. usque ad novemdecim compositus, & sic deinceps.” ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
tribus, &c. usque ad novemdecim compositus, & sic deinceps.”
ON VU’S THIN BASIS THEOREM IN WARING’S PROBLEM
"... V. Vu has recently shown that when k ≥ 2 and s is sufficiently large in terms of k, then there exists a set X(k), whose number of elements up to t is smaller than a constant times (t log t) 1/s, for which all large integers n are represented as the sum of s kth powers of elements of X(k) in order lo ..."
Abstract
 Add to MetaCart
V. Vu has recently shown that when k ≥ 2 and s is sufficiently large in terms of k, then there exists a set X(k), whose number of elements up to t is smaller than a constant times (t log t) 1/s, for which all large integers n are represented as the sum of s kth powers of elements of X(k) in order log n ways. We establish this conclusion with s ∼ k log k, improving on the constraint implicit in Vu’s work which forces s to be as large as k 4 8 k. Indeed, the methods of this paper show, roughly speaking, that whenever existing methods permit one to show that all large integers are the sum of H(k) kth powers of natural numbers, then H(k) + 2 variables suffice to obtain a corresponding conclusion for “thin sets, ” in the sense of Vu. 1.
Diophantine Methods for Exponential Sums, and Exponential Sums for Diophantine Problems
, 2003
"... Recent developments in the theory and application of the HardyLittlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developme ..."
Abstract
 Add to MetaCart
Recent developments in the theory and application of the HardyLittlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and CauchySchwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
THE HASSE PRINCIPLE FOR PAIRS OF DIAGONAL CUBIC FORMS
, 710
"... Abstract. By means of the HardyLittlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables. 1. Introduction. Early work of Lewis [14] and Birch [3, 4 ..."
Abstract
 Add to MetaCart
Abstract. By means of the HardyLittlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables. 1. Introduction. Early work of Lewis [14] and Birch [3, 4], now almost a halfcentury old, shows that pairs of quite general homogeneous cubic equations possess nontrivial integral solutions whenever the dimension of the corresponding intersection is suitably large (modern refinements have reduced this permissible affine dimension to 826; see [13]). When s is a natural number, let aj, bj (1 ≤ j ≤ s)