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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 ..."
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: 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. ..."
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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.
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 ..."
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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)
Waring’s Problem: A Survey
"... tribus, &c. usque ad novemdecim compositus, & sic deinceps.” ..."
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tribus, &c. usque ad novemdecim compositus, & sic deinceps.”
THE HASSE PRINCIPLE FOR SYSTEMS OF DIAGONAL CUBIC FORMS
"... Abstract. We establish the Hasse Principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r × r submatrix, thereby achieving the theoretical limit of the circle method for such systems. 1. ..."
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Abstract. We establish the Hasse Principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r × r submatrix, thereby achieving the theoretical limit of the circle method for such systems. 1.
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 ..."
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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 ..."
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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.
ON SIMULTANEOUS DIAGONAL INEQUALITIES
"... Let F1,..., Ft be diagonal forms of degree k with real coefficients in s variables, and let τ be a positive real number. The solubility of the system of inequalities F1(x)  < τ,..., Ft(x)  < τ in integers x1,..., xs has been considered by a number of authors over the last quartercentury, ..."
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Let F1,..., Ft be diagonal forms of degree k with real coefficients in s variables, and let τ be a positive real number. The solubility of the system of inequalities F1(x)  < τ,..., Ft(x)  < τ in integers x1,..., xs has been considered by a number of authors over the last quartercentury,