On Small Solutions to Quadratic Congruences

2010/04/05 by Igor Shparlinski, Shparlinski, Igor
Mathematics · #11D79 #11J71 #11L07 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D79 #msc:11J71 #msc:11L07

paper · pdf · doi:10.48550/arxiv.1004.0715

arxiv created 2010/04/05 · arxiv updated 2010/04/07

Abstract

We estimate the deviation of the number of solutions of the congruence m²-n² ≡ c \pmod q, 1 ≤ m ≤ M, 1≤ n ≤ N, from its expected value on average over c=1, ..., q. This estimate is motivated by the recently established by D. R. Heath-Brown connection between the distibution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function αk², k=1,2,... with a real α.

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