Far Eastern Mathematical Journal

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On the distribution of points on the modular hyperbola

V. I. Bernik, A. V. Ustinov

2014, issue 2, P. 141–155

Abstract

It is known that solutions of determinant equation $\bigl|\begin{smallmatrix} a & x\\ y & z \end{smallmatrix}\bigr|=q$ are uniformly distributed. The article contains similar results when the variables satisfy additional restrictions $(a,x)=1$ or $(a,x,y,z)=1$.

Keywords: Kloosterman sums, modular hyperbola

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References

[1] Vinogradov A. I., “Ukorochennoe uravnenie dlya svertok”, Zapiski nauchnyx seminarov POMI, 211, Nauka, 1994, 104–119 . [2] Linnik Yu. V., Skubenko B. F., “Asimptoticheskoe raspredelenie celochislennyx matric tret'ego poryadka”, Vestnik LGU, ser. mat., mex., astronom., 3:13 (1964), 25–36. [3] Ustinov A. V., “O chisle reshenij sravneniya $xy\equiv \ell\pmod{q}$ pod grafikom dvazhdy nepreryvno differenciruemoj funkcii”, Algebra i analiz, 20:5 (2008), 186–216. [4] Ustinov A. V., “Reshenie zadachi Arnol'da o slaboj asimptotike dlya chisel Frobeniusa s tremya argumentami”, Mat. sbornik, 200:4 (2009), 131–160. [5] Ustinov A. V., “O raspredelenii reshenij determinantnogo uravneniya”, Matematicheskij sbornik (v pechati). [6] Andersson J., Summation formulae and zeta functions, Stockholm University, Faculty of Science, Department of Mathematics, 2006. [7] Estermann T., “On {K}loosterman's sum”, Mathematika, 8 (1961), 83–86. [8] Hardy G. H., Write E. M., An Introduction to the Number Theory, Clarendon Press, Oxford, 1979. [9] Shparlinski I. E., “Modular hyperbolas”, Jpn. J. Math., 7 (2012), 235–294. [10] Weil A., “On some exponential sums”, Proc. Natl. Acad. Sci. USA, 34 (1948), 204–207.

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