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

Alexandru Zaharescu

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 3, page 863-872
  • ISSN: 1246-7405

Abstract

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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 ∈ 𝔽 p : r ˜ ( x ) > r ˜ ( x + 1 ) , where r ˜ ( x ) and r ˜ ( x + 1 ) denote the least positive representatives of r ( x ) and r ( x + 1 ) in ℤ modulo p ℤ .

How to cite

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Zaharescu, Alexandru. "The distribution of the values of a rational function modulo a big prime." Journal de théorie des nombres de Bordeaux 15.3 (2003): 863-872. <http://eudml.org/doc/249072>.

@article{Zaharescu2003,
abstract = {Given a large prime number $p$ and a rational function $r(X)$ defined over $\mathbb \{F\}_p = \mathbb \{Z\} / p \mathbb \{Z\}$, we investigate the size of the set $\left\lbrace x \in \mathbb \{F\}_p : \tilde\{r\}(x) &gt; \tilde\{r\}(x +1)\right.$, where $\tilde\{r\}(x)$ and $\tilde\{r\}(x + 1)$ denote the least positive representatives of $r(x)$ and $r(x+1)$ in $\mathbb \{Z\}$ modulo $p\mathbb \{Z\}$.},
author = {Zaharescu, Alexandru},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {rational function; distribution of values},
language = {eng},
number = {3},
pages = {863-872},
publisher = {Université Bordeaux I},
title = {The distribution of the values of a rational function modulo a big prime},
url = {http://eudml.org/doc/249072},
volume = {15},
year = {2003},
}

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
PB - Université Bordeaux I
VL - 15
IS - 3
SP - 863
EP - 872
AB - Given a large prime number $p$ and a rational function $r(X)$ defined over $\mathbb {F}_p = \mathbb {Z} / p \mathbb {Z}$, we investigate the size of the set $\left\lbrace x \in \mathbb {F}_p : \tilde{r}(x) &gt; \tilde{r}(x +1)\right.$, where $\tilde{r}(x)$ and $\tilde{r}(x + 1)$ denote the least positive representatives of $r(x)$ and $r(x+1)$ in $\mathbb {Z}$ modulo $p\mathbb {Z}$.
LA - eng
KW - rational function; distribution of values
UR - http://eudml.org/doc/249072
ER -

References

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  1. [1] E. Bombieri, On exponential sums in finite fields. Amer. J. of Math.88 (1966), 71-105. Zbl0171.41504MR200267
  2. [2] C. Cobeli, A. Zaharescu, Generalization of a problem of Lehmer. Manuscripta Math.104 no. 3 (2001), 301-307. Zbl1034.11042MR1828876
  3. [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. Zbl1025.11021MR1845688
  4. [4] R.K. Guy, Unsolved Problems in Number Theory. Springer-Verlag, New York - Berlin, 1981, (second edition 1994). Zbl0474.10001MR1299330
  5. [5] B.Z. Moroz, The distribution of power residues and non-residues. Vestnik LGU, 16 no. 19 (1961), 164-169. Zbl0109.03002MR140508
  6. [6] G.I. Perel'muter, On certain character sums. Uspechi Matem. Nauk, 18 (1963), 145-149. Zbl0146.05202MR148630
  7. [7] A. Weil, On some exponential sums. Proc Nat. Acad. Sci. U.S.A.34 (1948), 204-207. Zbl0032.26102MR27006
  8. [8] W. Zhang, On a problem of D. H. Lehmer and its generalization. Compositio Math.86 no. 3 (1993), 307-316. Zbl0783.11002MR1219630
  9. [9] W. Zhang, A problem of D. H. Lehmer and its generalization II. Compositio Math.91 no. 1 (1994), 47-56. Zbl0798.11001MR1273925
  10. [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. Zbl0826.11003MR1308126
  11. [11] W. Zhang, On the distribution of inverses modulo n. J. Number Theory61 no. 2 (1996), 301-310. Zbl0874.11006MR1423056
  12. [12] Z. Zheng, The distribution of Zeros of an Irreducible Curve over a Finite Field. J. Number Theory59 no. 1 (1996), 106-118. Zbl0862.11041MR1399701

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