The distribution of the values of a rational function modulo a big prime
Journal de théorie des nombres de Bordeaux, Volume 15 (2003) no. 3, pp. 863-872.

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

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

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author = {Alexandru Zaharescu},
title = {The distribution of the values of a rational function modulo a big prime},
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Alexandru Zaharescu. The distribution of the values of a rational function modulo a big prime. Journal de théorie des nombres de Bordeaux, Volume 15 (2003) no. 3, pp. 863-872. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_3_863_0/

[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