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Almost-primes represented by quadratic polynomials. (English) Zbl 1256.11049
Let \(q(x)\in\mathbb{Z}[x]\) be an irreducible quadratic polynomial with positive leading term, and suppose that for every prime \(p\) there is an integer \(n_p\) such that \(p\) does not divide \(q(n_p)\). Then it is shown that \(q(n)\) takes infinitely many \(P_2\) values, that is to say that \(\Omega(q(n))\leq 2\) for infinitely many integers \(n\). This result was stated by H. Iwaniec [Invent. Math. 47, 171–188 (1978; Zbl 0389.10031)], but the proof was given only for the case \(q(x)=x^2+1\). The present paper now gives the necessary details for the general case, following the sieve procedure used by Iwaniec.
Much of the argument is similar to that developed by Iwaniec, and the most significant new work is required in analysing the equidistribution modulo \(m\) of the roots of the congruence \(q(n)\equiv 0\pmod{m}\). This eventually depends, as in Iwaniec’s work, on Weil’s bound for the Kloosterman sum.

MSC:
11N36 Applications of sieve methods
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
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