Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Primes in the intervals between primes squared   The set of short intervals between consecutive primes squared has the
pleasant---but seemingly unexploited---property that each interval
$s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes.
Here we take advantage of this essential characteristic and present evidence
for the conjecture that $\pi_k \sim |s_k|/ \log p_{k+1}^2$, where $\pi_k$ is
the number of primes in $s_k$; or even stricter, that $y=x^{1/2}$ is both
necessary and sufficient for the prime number theorem to be valid in intervals
of length $y$. In addition, we propose and substantiate that the prime counting
function $\pi(x)$ is best understood as a sum of correlated random variables
$\pi_k$. Under this assumption, we derive the theoretical variance of
$\pi(p_{k+1}^2)=\sum_{j=1}^k \pi_j$, from which we are led to conjecture that
$|\pi({x})-\textrm{li}(x)| =O(\sqrt{\textrm{li}(x)})$. Emerging from our
investigations is the view that the intervals between consecutive primes
squared hold the key to a furthered understanding of the distribution of
primes; as evidenced, this perspective also builds strong support in favour of
the Riemann hypothesis.

0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga