Eine Bemerkung zur Franelsumme von Folgen. (Note on the Franel sum of sequences) (Q1114760)

For any \(N\in {\mathbb{N}}\) and \((\rho_ k)_ k\subseteq [0,1)\) let \((\rho_{\nu,N})_{\nu \leq N}\) denote the finite sequence, ordered according to their size, of the first N terms of \((\rho_ k)_ k\). We investigate the order of magnitude, for \(N\to \infty\), of \(\Delta_ N(\rho):=\sum_{\nu \leq N}(\rho_{\nu,N}-\nu /N)^ 2\) in connection with the discrepancy \(D_ n(\rho)\). Further, it is shown that for \((\sigma_ k)_ k\) the Farey sequence the known good upper estimate of \(\Delta_ N(\sigma)\) for a certain subsequence of N's (namely \(O(N^{- 1/2+\epsilon})\) under RH) does not hold for all N: We have \(\Delta_ N(\sigma)=\Omega (1)\).