We prove new results in sieve theory—an active subfield of number theory. Sieve theory constitutes a set of tools that are used to detect integers in sparse sets with a bounded number of prime factors.
After giving a short introduction to sieve theory, we prove an explicit form of the linear sieve, which corrects and improves upon the work of [Reference Bordignon1, Reference Nathanson13]. Using our explicit sieve, we give the following two applications related to longstanding conjectures of Legendre and Goldbach. Here and throughout, we assume that prime factors are counted with multiplicity.
Theorem 1 (See also [Reference Dudek and Johnston6]).
For all
$n\geq 1$
, there exists a number between
$n^2$
and
$(n+1)^2$
with at most
$4$
prime factors.
Theorem 2. Assuming the generalised Riemann hypothesis, every even
$n\geq 4$
can be written as the sum of a prime and a number with at most
$16$
prime factors.
Theorem 1 is the first result of this form to hold for all
$n\geq 1$
, and makes use of a recent large computation by Sorenson and Webster [Reference Sorenson and Webster15], along with an explicit version of Kuhn’s weighted sieve [Reference Kuhn11].
Theorem 2 simplifies and improves upon results from [Reference Hathi and Johnston8, Reference Johnston and Starichkova9]. Most notably, in [Reference Johnston and Starichkova9, Theorem 1.4], the author and Starichkova obtain the same result as Theorem 2 but with
$31$
prime factors instead of
$16$
. It is also possible to obtain a worse result that does not depend on the generalised Riemann hypothesis (see [Reference Bordignon, Johnston and Starichkova2, Theorem 1.6] and [Reference Johnston and Starichkova9, Theorem 1.3]), but we do not do so to limit scope.
Finally, we give a new asymptotic application of sieve theory, which generalises a series of Goldbach-like results. On this topic, our main theorem is as follows.
Theorem 3 (See also [Reference Johnston and Thomas10, Theorem 1.5]).
Let
$k\geq 2$
. There exists an integer
$M(k)$
such that every sufficiently large integer N can be expressed as
$$ \begin{align} N=p^k+\eta, \end{align} $$
where p is prime and
$\eta>0$
has at most
$M(k)$
prime factors. Here,
$M(k)=6k$
is admissible for all even
$k\geq 2$
and
$M(k)=4k$
is admissible for all odd
$k\geq 3$
. In addition, for sufficiently large
$k\geq k_{\varepsilon }$
, one can set
$$ \begin{align*} M(k)=(2+\varepsilon)k \end{align*} $$
for any
$\varepsilon>0$
. Or, assuming of the Elliott–Halberstam conjecture,
$$ \begin{align*} M(k)=(1+\varepsilon)k. \end{align*} $$
Our proof of Theorem 3 combines results for kth power residues with the weighted sieve method of Diamond et al. [Reference Diamond, Halberstam and Galway5]. Notably, for fixed
$k\geq 2$
, our results are the first of the form (1) for sufficiently large N with
$\eta $
having a uniformly bounded number of prime factors. For the simpler case
$k=1$
, one can represent large N of the form (1) with
$\eta $
having at most
$3$
prime factors, as a result of the work of Chen [Reference Chen3, Reference Chen4] and Li [Reference Li12].
We also prove a more technical version of Theorem 3 (see [Reference Johnston and Thomas10, Theorem 1.6]), which accounts for congruence conditions on N and improves upon classical work of Erdős [Reference Erdős7] and Rao [Reference Rao14].
