Exponential Sums with Farey Fractions

Igor E. Shparlinski¹

¹Department of Computing Macquarie University Sydney, NSW 2109, Australia

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 2, pp. 101-107

Résumé

For positive integers $m$ and $N$, we estimate the rational exponential sums with denominator $m$ over the reductions modulo $m$ of elements of the set $$ {\cal F}(N) = \{ s/r : r,s \in {\mathbb Z},\, \gcd(r,s) = 1, \, N\ge r > s \ge 1\} $$ of Farey fractions of order $N$ (only fractions $s/r$ with $\gcd(r,m)=1$ are considered).

Détail

DOI : 10.4064/ba57-2-2

Keywords: positive integers estimate rational exponential sums denominator reductions modulo elements set cal mathbb gcd farey fractions order only fractions gcd considered