Quadratic polynomials at prime arguments

Abstract

For a fixed quadratic irreducible polynomial f with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes p such that f(p) has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of n, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.

Appendix: Composition of two linear sieves

In this appendix, we formulate a reduction of Friedlander and Iwaniec on the composition of two sieves. In particular, we focus on linear sieves. The original statement with proof can be found in [4, Appendix A] or [5, Section 5.10].

Theorem 6.1

Let $(\lambda _1),$ $(\lambda _2)$ be two upper-bound linear sieves of levels $D_1,D_2,$ respectively, and let $g_1,g_2$ be density functions satisfying linear sieve conditions and

$$\begin{aligned} 0\leqslant g_1(p),g_2(p)\leqslant \tfrac{1}{2} \end{aligned}$$

(6.1)

for each prime p. Then

$$\begin{aligned} \left| \mathop {\sum \sum }_{(d_1,d_2)=1}\lambda _1(d_1)\lambda _2(d_2)g_1(d_1)g_2(d_2)\right| \leqslant \frac{4H\{1+o(1)\}}{\log D_1\log D_2} \end{aligned}$$

with

$$\begin{aligned} H := \prod _{p}(1-g_1(p)-g_2(p))(1-1/p)^{-2}. \end{aligned}$$

One can see that we have an extra condition (6.1) compared to the original version of Friedlander and Iwaniec, for whom the constant H should be replaced by the following larger one

$$\begin{aligned} \prod _{p}(1-g_1(p)-g_2(p)+2g_1(p)g_2(p))(1-1/p)^{-2}. \end{aligned}$$

The modification here is due to avoiding the use of the trivial inequality

$$\begin{aligned} \left| 1-\frac{g_1g_2}{(1-g_1)(1-g_2)}\right| \leqslant 1+\frac{g_1g_2}{(1-g_1)(1-g_2)} \end{aligned}$$

at primes.