Kloosterman's uniformly distributed sequence. (Q1415363)

Applying the theory of uniform distribution, especially the Erdős-Turán-Koksma inequality and the Koksma-Hlawka inequality, to the two-dimensional sequence \((a_{j}/n, a^{*}_{j}/n), j = 1,2\ldots, \varphi(n)\) (where \(a_{j}a^{*}_{j}\equiv 1 \pmod n,\, a_{j}, a^{*}_{j}\in[1,n] \text{ and }\varphi(n)\) is the Euler function) the authors prove an upper bound for the discrepancy \(D^{*}_{\varphi (n)}\) of this sequence: \[ D^{*}_{\varphi(n)}((a_{j}/n,a^{*}_{j}/ n))=O(d(n)\sqrt{n}(\log \varphi (n))^{2}/\varphi(n)) \] These results improve and unify some of \textit{W. Zhang's} results [J. Number Theory 52, 1--6 (1995; Zbl 0826.11002); J. Number Theory 61, 301--310 (1996; Zbl 0874.11006); Acta Math. Hung. 76, 17--30 (1997; Zbl 0906.11043)].