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 1838072 | Zbl 1046.11068

BibTeX
RIS

@article{JTNB_2001__13_1_93_0,
     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},
     zbl = {1046.11068},
     mrnumber = {1838072},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_93_0/}
}

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AU  - Coppola, Giovanni
AU  - Salerno, Saverio
TI  - On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2001
DA  - 2001///
SP  - 93
EP  - 102
VL  - 13
IS  - 1
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_2001__13_1_93_0/
UR  - https://zbmath.org/?q=an%3A1046.11068
UR  - https://www.ams.org/mathscinet-getitem?mr=1838072
LA  - en
ID  - JTNB_2001__13_1_93_0
ER  -

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/

Bibliographie
Cité par

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