Results 1  10
of
14
Prime and Almost Prime Integral Points on Principal Homogeneous Spaces
"... We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.
BOUNDED GAPS BETWEEN PRODUCTS OF PRIMES WITH APPLICATIONS TO IDEAL CLASS GROUPS AND ELLIPTIC CURVES
"... and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
and Y ld r m use a variant of the Selberg sieve to prove the existence of small gaps between E2 numbers, that is, squarefree numbers with exactly two prime factors. We apply their techniques to prove similar bounds for Er numbers for any r ≥ 2, where these numbers are required to have all of their prime factors in a set of primes P. Our result holds for any P of positive density that satis es a SiegelWal sz condition regarding distribution in arithmetic progressions. We also prove a stronger result in the case that P satis es a BombieriVinogradov condition. We were motivated to prove these generalizations because of recent results of Ono [22] and Soundararajan [25]. These generalizations yield applications to divisibility of class numbers, nonvanishing of critical values of Lfunctions, and triviality of ranks of elliptic curves. 1.
Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
, 2008
"... ..."
(Show Context)
, (2)
"... Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture. ..."
Abstract
 Add to MetaCart
Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture.
Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
, 2008
"... ..."
(Show Context)
J ( ω) = Π ( P−1 − χ ( P)), (2) 2
"... Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture. ..."
Abstract
 Add to MetaCart
Using Jiang function we prove Jiang prime ktuple theorem. We prove that the HardyLittlewood prime ktuple conjecture is false. Jiang prime ktuple theorem can replace the HardyLittlewood prime ktuple conjecture.
1 A REMARK ON THE GEOMETRY OF SPACES OF FUNCTIONS WITH PRIME FREQUENCIES.
"... Abstract. For any positive integer r, denote by Pr the set of all integers γ ∈ Z having at most r prime divisors. We show CPr (T), the space of all continuous functions on the circle T whose Fourier spectrum lies in Pr, contains a complemented copy of ℓ1. In particular, CPr (T) is not isomorphic to ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. For any positive integer r, denote by Pr the set of all integers γ ∈ Z having at most r prime divisors. We show CPr (T), the space of all continuous functions on the circle T whose Fourier spectrum lies in Pr, contains a complemented copy of ℓ1. In particular, CPr (T) is not isomorphic to C(T), nor to the disc algebra A(D). A similar result holds in the L1 setting.
Theorem 1.
"... 2)2()1() ( =+=+ = xdxdxd infinitelyoften. （1） where)(xd represents the number of distinct prime factors of x, 1)3(,1) ( =Σ = dxd xP 2)15 ( =d, 3)105 ( =d. Proof (see[1] p.146 theorem 3.1.154). Prime equations are 16,215,110 141312 +=+=+ = pppppp （2） ..."
Abstract
 Add to MetaCart
2)2()1() ( =+=+ = xdxdxd infinitelyoften. （1） where)(xd represents the number of distinct prime factors of x, 1)3(,1) ( =Σ = dxd xP 2)15 ( =d, 3)105 ( =d. Proof (see[1] p.146 theorem 3.1.154). Prime equations are 16,215,110 141312 +=+=+ = pppppp （2）