4 research outputs found
Strings of congruent primes in short intervals II
Let be the sequence of all primes. Let be
an arbitrarily small but fixed positive number, and fix a coprime pair of
integers and . We will establish a lower bound for the number of
primes , up to , such that both
and simultaneously hold. As a lower
bound for the number of primes satisfying the latter condition, the bound we
obtain improves upon a bound obtained by D. Shiu
The existence of small prime gaps in subsets of the integers
We consider the problem of finding small prime gaps in various sets of
integers . Following the work of Goldston-Pintz-Yildirim, we will
consider collections of natural numbers that are well-controlled in arithmetic
progressions. Letting denote the -th prime in , we will
establish that for any small constant , the set constitutes a positive proportion of
all prime numbers. Using the techniques developed by Maynard and Tao we will
also demonstrate that has bounded prime gaps. Specific examples,
such as the case where is an arithmetic progression have already
been studied and so the purpose of this paper is to present results for general
classes of sets
Dense clusters of primes in subsets
We prove a generalization of the author's work to show that any subset of the
primes which is `well-distributed' in arithmetic progressions contains many
primes which are close together. Moreover, our bounds hold with some uniformity
in the parameters. As applications, we show there are infinitely many intervals
of length containing primes,
and show lower bounds of the correct order of magnitude for the number of
strings of congruent primes with .Comment: 35 pages; clarified some statement