On the least almost-prime in arithmetic progression

TL;DR

This paper improves the upper bound on the least almost-prime with at most two prime factors in an arithmetic progression, showing it is bounded by a constant times q^{1.8345} for large q, refining previous results.

Contribution

It establishes a tighter upper bound on the least almost-prime in arithmetic progression, improving upon Iwaniec's earlier bound of q^{1.845}.

Findings

Bound on least almost-prime:  q^{1.8345} for large q.

Improvement over previous bound of q^{1.845}.

Advances understanding of distribution of almost-primes in arithmetic progressions.

Abstract

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least almost-prime $\mathcal{P}_2$ which satisfies $\mathcal{P}_2\equiv a\pmod q$. In this paper, it is proved that for sufficiently large $q$, there holds \begin{equation*} \mathcal{P}_2(a,q)\ll q^{1.8345}. \end{equation*} This result constitutes an improvement upon that of Iwaniec, who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$.