On the uniform distribution (mod 1) of the Farey fractions and \(\ell ^ p\) spaces
We estimate the discrepancy \[ E_ f(N)=\sum _{q\leq N}\sum ^{*qa=1}f(a/q)-(\sum _{q\leq N}\sum ^{*qa=1}1)\int ^{1}_{0}f(x) dx \] for a rather wide class of functions f. If f is absolutely continuous and \(f'\in L^ p[0,1]\), and if we assume the \(\alpha\)-quasi Riemann hypothesis, then we have for every \(\epsilon >0\) \[ E_ f(N)=O(N^{\max (\alpha,1/p)+\epsilon}),\quad E_ f(N)=\Omega (N^{\max (\alpha,1/p)-\epsilon}). \] Define the ''deviation coefficients'' D(n) by \[ D(n)/n = (1/n)\sum ^{n}_{a=1}f(a/n)-\int ^{1}_{0}f(x) dx. \] Then we have \[ (*)\quad D(n)=-(1/2\pi)\sum ^{\infty}_{k=1}c(kn)/k, \] where (c(n)) are the Fourier coefficients of f'. In view of the Hausdorff-Young theorem, we also study the structure of the linear transformation (c(n))\(\to (D(n))\) given by (*) and obtain the best possible result that if \((c(n))\in \ell ^ p\) with \(p\geq 1\), then \((D(n))\in \ell ^ r\) for every \(r>p\).
scientific article; zbMATH DE number 3979179