Some problems of diophantine approximation in the theory of the Riemann zeta function (Q1201497)

The author investigates the distribution of \(\{{{\alpha\gamma} \over {2\pi}}\}\), where \(\alpha>0\) is a given number, \(\gamma\) denotes positive ordinates of the zeros of \(\zeta(s)\), and \(\{x\}\) is the fractional part of \(x\). If, as usual, \(N(T)\) denotes the number of \(\gamma\)'s such that \(\gamma\leq T\), then it is proved that \[ {\textstyle{1\over {N(T)}}} \sum_{\gamma\leq T} \Bigl(\bigl\{ {\textstyle {{\alpha\gamma} \over {2\pi}}\bigr\}-{1\over 2}} \Bigr)\;\ll\;R(T), \tag{1} \] where unconditionally \(R(T)= (\log \log T/\log T)^{1/2}\), and \(R(T)= (\log T)^{\varepsilon-1}\) if the Riemann hypothesis holds. The author also investigates the sum in (1) when the quantity in brackets is replaced by \(\{ {{\alpha\gamma} \over {2\pi}}\}^ 2- \{ {{\alpha\gamma} \over {2\pi}}\}+ {1\over 6}\). The proofs depend on the author's bounds for sums of \(X^{i\gamma}\) and the number of \(\gamma\)'s up to \(T\) for which \(0\leq \{ {{\alpha\gamma} \over {2\pi}}\}\leq \beta\) for a given \(0\leq\beta\leq 1\).