On the Euler function on differences between the coordinates of points on modular hyperbolas

• Autorzy: Shparlinski I. E.
• Języki publikacji: EN

Abstrakty

EN

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola Ηa,p(X,Y) = {(x,y) : x,y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. We give asymptotic formulas for the average values (x,y)∈ ... [wzór] with the Euler function φ(k) on the difference between the components of points of Ηa,p(X,Y).

Słowa kluczowe

EN

modular hyperbola; Euler function; discrepancy

PL

hiperbola modularna; funkcja Eulera

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: Vol. 56, no 1
• Strony: 1--7
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Shparlinski I. E.

• Department of Computing, Macquarie University, North Ryde, NSW 2109, Australia,

Bibliografia

• [1] M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Springer, Berlin, 1997.
• [2] K. Ford, M. R. Khan, I. E. Shparlinski and C. L. Yankov, On the maximal difference between an element and its inverse in residue rings, Proc. Amer. Math. Soc. 133 (2005), 3463-3468.
• [3] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004.
• [4] M. R. Khan and I. E. Shparlinski, On the maximal difference between an element and its inverse modulo n, Period. Math. Hungar. 47 (2003), 111—117.
• [5] I. E. Shparlinski, Distribution of inverses and multiples of small integers and the Sato-Tate conjecture on average, Michigan Math. J., to appear.
• [6] I. E. Shparlinski and A. Winterhof, Distances between the points on modular hyperbolas, J. Number Theory, to appear.