Results 1  10
of
11
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 4 (1 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.
Small
"... gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers ..."
Abstract
 Add to MetaCart
gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
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.
, (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.
unknown title
, 803
"... Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers ..."
Abstract
 Add to MetaCart
Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers
� � � P, � ( P)
"... Using Jiang function we prove Jiang prime ktuple theorem.We find true singular series. Using the examples we prove the HardyLittlewood prime ktuple conjecture with wrong singular series.. Jiang prime ktuple theorem will replace the HardyLittlewood prime ktuple conjecture. (A) Jiang prime ktup ..."
Abstract
 Add to MetaCart
Using Jiang function we prove Jiang prime ktuple theorem.We find true singular series. Using the examples we prove the HardyLittlewood prime ktuple conjecture with wrong singular series.. Jiang prime ktuple theorem will replace the HardyLittlewood prime ktuple conjecture. (A) Jiang prime ktuple theorem with true singular series[1, 2]. We define the prime ktuple equation p, p � ni, (1) where 2 n, i �1, � k�1. i we have Jiang function [1, 2]
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
 Add to MetaCart
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.