On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 93-102.

Ce texte donne de nouveaux résultats sur la répartition dans les progressions arithmétiques (modulo un produit de deux nombres premiers) des valeurs (an+b)(cn+d) prises par un polynôme quadratique réductible lorsque l’entier n varie dans des intervalles courts n∈[x,x+x ϑ ], où ϑ∈(0,1]. Nous utilisons ici la méthode de dispersion, pour obtenir un niveau de répartition au delà du niveau classique θ. Nous obtenons pour niveau 3ϑ/2, améliorant en cela la valeur 3ϑ-3/2 obtenue par le grand crible. Nous terminons par une comparaison graphique des deux approches.

This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials (an+b)(cn+d) in short intervals n∈[x,x+x ϑ ], where now ϑ∈(0,1]. Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level ϑ, reaching 3ϑ/2 (thus improving also the level of the previous paper, i.e. 3ϑ-3/2), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

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Coppola, Giovanni; Salerno, Saverio. On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 93-102. http://www.numdam.org/item/JTNB_2001__13_1_93_0/

[1] E. Bombieri, Le grand crible dans la théorie analytique des nombres. Astérisque 18, Société mathématique de France, 1974. | MR | Zbl

[2] G. Coppola, S. Salerno, On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals. Functiones et Approximatio XXVIII (2001), 75-81. | MR | Zbl

[3] H. Halberstam, H.E. Richert, Sieve methods. Academic Press, London, 1974. | MR | Zbl

[4] C. Hooley, On the greatest prime factor of a quadratic polynomial. Acta Math. 117 (1967), 281-299. | MR | Zbl

[5] H. Iwaniec, Almost-primes represented by quadratic polynomials. Inventiones Math. 47 (1978), 171-188. | MR | Zbl

[6] Ju.V. Linnik, The dispersion method in binary additive problems. American Mathematical Society, 1963. | MR | Zbl

[7] S. Salerno, A. Vitolo, On the distribution in the arithmetic progressions of reducible quadratic polynomials. Izwestiya Rossiyskoy AN, Math.Series 58 (1994), 211-223. | MR | Zbl