## 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

### Citations:

Zbl 0509.10029; Zbl 0489.10038; Zbl 0502.10021
Full Text:

### 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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