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The largest prime factor of \(X^3+2\). (English) Zbl 1023.11048
Let \(f\) be an irreducible polynomial with positive leading coefficient, and define \(P(x;f)\) to be the largest prime divisor of \(\prod_{n\leq x} f(n)\). C. Hooley [J. Reine Angew. Math. 303/304, 21–50 (1978; Zbl 0391.10028)] gave a proof that \(P(x,X^3+2) > x^{31/30}\) provided one assumes “Hypothesis \(R^*\)”, a best possible estimate for short Ramanujan-Kloosterman sums. In this paper, the author proves the unconditional estimate \[ P(x,X^3+2) > x^{1+\varpi}\quad\text{with}\quad \varpi=10^{-303}. \] Although the constant \(\varpi\) is very small, this result is significant because it is the first unconditional result of the form \(P(x,f)> x^{1+\delta}\) for a polynomial of \(f\) of degree exceeding \(2\).
Problems involving \(P(x,f)\) were first considered by Chebyshev, who sketched a proof that \(P(x,X^2+1)/x\to \infty\). In this paper, the author develops a novel variant of Chebyshev’s method; this variant method is quite likely to be useful for other problems of Chebyshev type. The original Chebyshev method requires non-trivial estimates of appropriate error terms summed over prime arguments. In the variant, the error terms are summed over “smooth” arguments; i.e., over numbers with no large prime factors, and one has considerable freedom about choosing the arguments. The resulting error terms in this paper are estimated via a \(q\)-analogue of van der Corput’s method. The basic principles of this method were first sketched in the author’s paper “Hybrid bounds for \(L\)-functions: a \(q\)-analogue of van der Corput’s method and a \(t\)-analogue of Burgess’ method”, Recent progress in analytic number theory, (ed. H. Halberstam and C. Hooley, Academic Press, London, Vol. 1 (Durham 1979)), 121–126 (1981; Zbl 0457.10021)].

MSC:
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11L07 Estimates on exponential sums
11N36 Applications of sieve methods
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