Geodesic
On Points of the Modular Hyperbola under the Graph of a Linear Function
Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 296-301.

Voir la notice de l'article provenant de la source Math-Net.Ru

For the number of solutions of the congruence $xy\equiv 1\,(\operatorname{mod} p)$ under the graph of a linear function, an asymptotic formula with square root cancellation in the error term is proved.
Keywords: modular hyperbola
Mots-clés : Fourier coefficient. 
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A. V. Ustinov. On Points of the Modular Hyperbola under the Graph of a Linear Function. Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 296-301. http://geodesic.mathdoc.fr/item/MZM_2015_97_2_a11/

[1] I. E. Shparlinski, “Modular hyperbolas”, Japan. J. Math., 7:2 (2012), 235–294 | DOI | MR | Zbl

[2] A. V. Ustinov, “O chisle reshenii sravneniya $xy\equiv l\,(\operatorname{mod}q)$ pod grafikom dvazhdy nepreryvno differentsiruemoi funktsii”, Algebra i analiz, 20:5 (2008), 186–216 | MR | Zbl

[3] A. Weil, “On some exponential sums”, Proc. Natl. Acad. Sci. USA, 34 (1948), 204–207 | DOI | MR | Zbl

[4] V. A. Bykovskii, “Asimptoticheskie svoistva tselykh tochek $(a_1,a_2)$, udovletvoryayuschikh sravneniyu $a_1a_2\equiv l(q)$”, Analiticheskaya teoriya chisel i teoriya funktsii. 4, Zap. nauchn. sem. LOMI, 112, Izd-vo «Nauka», Leningrad. otd., L., 1981, 5–25 | MR | Zbl

[5] O. A. Gorkusha, “O konechnykh tsepnykh drobyakh spetsialnogo vida”, Chebyshevskii sb., 9:1 (2008), 80–107 | MR | Zbl

[6] A. V. Ustinov, “O raspredelenii chisel Frobeniusa s tremya argumentami”, Izv. RAN. Ser. matem., 74:5 (2010), 145–170 | DOI | MR | Zbl

[7] I. E. Shparlinski, A. Winterhof, “On the number of distances between the coordinates of points on modular hyperbolas”, J. Number Theory, 128:5 (2008), 1224–1230 | DOI | MR | Zbl

[8] A. V. Ustinov, “Spinovye tsepochki i zadacha Arnolda o statistikakh Gaussa–Kuzmina dlya kvadratichnykh irratsionalnostei”, Matem. sb., 204:5 (2013), 143–160 | DOI | MR | Zbl

[9] C. Cobeli, Y. Gallot, P. Moree, A. Zaharescu, “Sister Beiter and Kloosterman: a tale of cyclotomic coefficients and modular inverses”, Indag. Math. (N.S.), 24:4 (2013), 915–929 | MR | Zbl