On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II

Coppola, Giovanni; Salerno, Saverio

Coppola, Giovanni ; Salerno, Saverio

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 93-102.

Résumé
Abstract

Ce texte donne de nouveaux résultats sur la répartition dans les progressions arithmétiques (modulo un produit de deux nombres premiers) des valeurs $(a n + b) (c n + d)$ prises par un polynôme quadratique réductible lorsque l’entier $n$ varie dans des intervalles courts $n \in [x, x + x^{ϑ}]$ , où $ϑ \in (0, 1]$ . Nous utilisons ici la méthode de dispersion, pour obtenir un niveau de répartition au delà du niveau classique $θ$ . Nous obtenons pour niveau $3 ϑ / 2$ , améliorant en cela la valeur $3 ϑ - 3 / 2$ 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 $(a n + b) (c n + d)$ in short intervals $n \in [x, x + x^{ϑ}]$ , where now $ϑ \in (0, 1]$ . Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level $ϑ$ , reaching $3 ϑ / 2$ (thus improving also the level of the previous paper, i.e. $3 ϑ - 3 / 2$ ), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

MR Zbl

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     author = {Coppola, Giovanni and Salerno, Saverio},
     title = {On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, {II}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {93--102},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     mrnumber = {1838072},
     zbl = {1046.11068},
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     url = {http://archive.numdam.org/item/JTNB_2001__13_1_93_0/}
}

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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://archive.numdam.org/item/JTNB_2001__13_1_93_0/

Bibliographie
Cité par

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