Almost-prime values of polynomials at prime arguments

Abstract

We consider almost‐primes of the form f(p) where f is an irreducible polynomial over double-struckZ and p runs over primes. We improve a result of Richert for polynomials of degree at least 3. In particular, we show that, when the degree is large, there are infinitely many primes p for which f(p) has at most degf+O(logdegf) prime factors.

Almost-prime values of reducible polynomials at prime arguments

“…for d | P (z) and p z large enough to ensure that p ∤ H(0) (see proof of Lemma 4.2 in [7]). The sieve dimension is g + 1 in this case since the density function ρ 2 (d)/d appearing in (12) satisfies ( 14)…”

“…In this paper, we adopt a sieve method developed by A. J. Irving in [7] to prove Theorem 1. Let H(n) = h 1 (n) · · · h g (n), where h i are distinct irreducible polynomials each with integer coefficients and deg h i = k for all i = 1, .…”

“…g \ k 1 2 3 4 5 6 7 8 9 The case g = 1 was first investigated by H.-E. Richert in 1969 [9], who showed that for each k 1, r = 2k + 1 is an admissible choice. Virtually no progress was made until Irving's work in 2015 [7], which showed that one could take an r of the form r = k + O(log k). Explicit bounds for the O-term, as well as explicit values for r when k is small, are available in [7] and [8].…”

Almost-prime polynomials at prime arguments

“…As one of the concluding remarks in [8], Irving suggested the possibility of improvements by applying the Diamond-Halberstam-Richert sieve in place of the the beta sieve. We carry out his suggestion and extend all sifting functions into their respective non-elementary ranges.…”

On a Diophantine inequality involving a prime and an almost-prime

“…Selberg's trick can often help us slightly expand the range of sifting, e.g. see [9], where the sifting set is naturally multiplicative by the Chinese reminder theorem, and thus is easier to handle. However, the sifting set here has no multiplicative structure, so we have to use other tricks to conquer.…”