The distribution of the values of a rational function modulo a big prime

Zaharescu, Alexandru

Zaharescu, Alexandru

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 863-872.

Résumé
Abstract

Étant donnés un grand nombre premier $p$ et une fonction rationnelle $r (X)$ définie sur $𝔽_{p} = ℤ / p ℤ$ , on évalue la grandeur de l’ensemble $\{x \in 𝔽_{p} : \tilde{r} (x) > \tilde{r} (x + 1)\}$ , où $\tilde{r} (x)$ et $\tilde{r} (x + 1)$ sont les plus petits représentants de $r (x)$ et $r (x + 1)$ dans $ℤ$ modulo $p ℤ$ .

Given a large prime number $p$ and a rational function $r (X)$ defined over $𝔽_{p} = ℤ / p ℤ$ , we investigate the size of the set $\{x \in 𝔽_{p} : \tilde{r} (x) > \tilde{r} (x + 1)$ , where $\tilde{r} (x)$ and $\tilde{r} (x + 1)$ denote the least positive representatives of $r (x)$ and $r (x + 1)$ in $ℤ$ modulo $p ℤ$ .

MR Zbl

@article{JTNB_2003__15_3_863_0,
     author = {Zaharescu, Alexandru},
     title = {The distribution of the values of a rational function modulo a big prime},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {863--872},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     mrnumber = {2142241},
     zbl = {02184629},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_2003__15_3_863_0/}
}

TY  - JOUR
AU  - Zaharescu, Alexandru
TI  - The distribution of the values of a rational function modulo a big prime
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2003
SP  - 863
EP  - 872
VL  - 15
IS  - 3
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_2003__15_3_863_0/
LA  - en
ID  - JTNB_2003__15_3_863_0
ER  -

%0 Journal Article
%A Zaharescu, Alexandru
%T The distribution of the values of a rational function modulo a big prime
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 863-872
%V 15
%N 3
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_2003__15_3_863_0/
%G en
%F JTNB_2003__15_3_863_0

Zaharescu, Alexandru. The distribution of the values of a rational function modulo a big prime. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 863-872. http://archive.numdam.org/item/JTNB_2003__15_3_863_0/

Bibliographie
Cité par

[1] E. Bombieri, On exponential sums in finite fields. Amer. J. of Math. 88 (1966), 71-105. | MR | Zbl

[2] C. Cobeli, A. Zaharescu, Generalization of a problem of Lehmer. Manuscripta Math. 104 no. 3 (2001), 301-307. | MR | Zbl

[3] C. Cobeli, A. Zaharescu, On the distribution of the Fp-points on an affine curve in r dimensions. Acta Arith. 99 no. 4 (2001), 321-329. | MR | Zbl

[4] R.K. Guy, Unsolved Problems in Number Theory. Springer-Verlag, New York - Berlin, 1981, (second edition 1994). | MR | Zbl

[5] B.Z. Moroz, The distribution of power residues and non-residues. Vestnik LGU, 16 no. 19 (1961), 164-169. | MR | Zbl

[6] G.I. Perel'Muter, On certain character sums. Uspechi Matem. Nauk, 18 (1963), 145-149. | MR | Zbl

[7] A. Weil, On some exponential sums. Proc Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. | MR | Zbl

[8] W. Zhang, On a problem of D. H. Lehmer and its generalization. Compositio Math. 86 no. 3 (1993), 307-316. | Numdam | MR | Zbl

[9] W. Zhang, A problem of D. H. Lehmer and its generalization II. Compositio Math. 91 no. 1 (1994), 47-56. | Numdam | MR | Zbl

[10] W. Zhang, On the difference between a D. H. Lehmer number and its inverse modulo q. Acta Arith. 68 no. 3 (1994), 255-263. | MR | Zbl

[11] W. Zhang, On the distribution of inverses modulo n. J. Number Theory 61 no. 2 (1996), 301-310. | MR | Zbl

[12] Z. Zheng, The distribution of Zeros of an Irreducible Curve over a Finite Field. J. Number Theory 59 no. 1 (1996), 106-118. | MR | Zbl