Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A sieve problem and its application   Let $\theta$ be an arithmetic function and let $\mathcal{B}$ be the set of
positive integers $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$, which satisfy
$p_{j+1} \le \theta ( p_1^{\alpha_1}\cdots p_{j}^{\alpha_{j}})$ for $0\le j <
k$. We show that $\mathcal{B}$ has a natural density, provide a criterion to
determine whether this density is positive, and give various estimates for the
counting function of $\mathcal{B}$. When $\theta(n)/n$ is non-decreasing, the
set $\mathcal{B}$ coincides with the set of integers $n$ whose divisors $1=d_1<
d_2 < \ldots <d_{\tau(n)}=n$ satisfy $d_{j+1} \le \theta( d_j )$ for $1\le j
<\tau(n)$.

17
The categories of this paper are: q-fin
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio