Abstract
In this paper, we consider the congruence equation [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the multiplicative inverse of [Formula: see text]. We prove that its number of solutions is asymptotic to [Formula: see text] when [Formula: see text] by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when [Formula: see text], the number of solutions has order of magnitude at least [Formula: see text]. It would be interesting to better understand its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet [Formula: see text]-functions at [Formula: see text].
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