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On the greatest prime divisor of quadratic sequences. (English) Zbl 0762.11030

In 1982 J.-M. Deshouillers and H. Iwaniec [Ann. Inst. Fourier 32, No. 4, 1-11 (1982; Zbl 0489.10038)] proved that, if \(P_ x\) denotes the greatest prime divisor of \(\prod_{n\leq x}(n^ 2+1)\) then for any \(\varepsilon>0\) \[ P_ x>x^{\gamma_ 0-\varepsilon}\quad\text{where } \gamma_ 0=1.2024\dots \] Assuming Selberg’s conjecture \(\gamma_ 0\) could be replaced by \(\gamma_ 1=\sqrt{3/2}=1.2247\dots\).
The article under review deals with \(P_ x(\beta,\Theta)\) which denotes the greatest prime divisor of \(\prod_{n\leq x}\prod_{q\in{\mathcal B},q\leq x^ \Theta}(n^ 2+q^ 2)\) where \({\mathcal B}\) is a rather thin set of primes, satisfying the density condition \(\text{card}\{b\in{\mathcal B} |\;b\leq y\}\geq y^ \beta\), \(y\to\infty\). Using the method of J.-M. Deshouillers and H. Iwaniec in [Invent. Math. 70, 219-288 (1982; Zbl 0502.10021)] for estimating multilinear forms in Kloosterman sums the author obtains for any \(\varepsilon>0\) the lower bound \[ P_ x(\beta,\Theta)>x^{\gamma-\varepsilon}, \;x\to\infty \] with some \(\gamma(\beta,\Theta)>0\) the expression for which is to involved to be given here. Yet it is worth mentioning that \(\gamma(\beta,\Theta)\to\gamma_ 0\) as \(\Theta\to 0\) while for any \(\delta>0\) one finds that \(\gamma(\beta,\sqrt{3/2}-1)\geq\gamma_ 1- \delta\) if \(|\beta-1|\) is sufficiently small.

MSC:

11N13 Primes in congruence classes
11L05 Gauss and Kloosterman sums; generalizations
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References:

[1] Deshouillers, J.-M. and Iwaniec, H., On the greatest prime factor of n2 + 1, Ann. Inst. Fourier, Grenoble32, 4(1982), 1-11. · Zbl 0489.10038
[2] Deshouillers, J.-M. and Iwaniec, H., Kloosterman sums and Fourier coefficients of cusp forms, Inv. math.70 (1982), 219-288. · Zbl 0502.10021
[3] Hooley, C., Applications of sieve methods to the theory of numbers, Cambridge University Press, London (1976). · Zbl 0327.10044
[4] Hooley, C., On the greatest prime factor of a quadratic polynomial, Acta Math.117 (1967), 281-299. · Zbl 0146.05704
[5] Iwaniec, H., Rosser’s sieve, Acta Arith.36 (1980), 171-202. · Zbl 0435.10029
[6] Smith, H.J.S., Report on the theory of numbers, Collected Mathematical Papers, vol. I, reprint, Chelsea, 1965.
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