On the distribution of reduced residues
Let \(P=\varphi (q)/q\) and write \[ M_ k(q;h)=\sum_{1\leq n\leq q}\biggl(\sum_{1\leq m\leq h;\;(m+n,q)=1}1-hP\biggr)^ k, \] the \(k\)th moment of the number of reduced residues mod \(q\) in an interval of length \(h\) about its mean \(hP\). After remarking that the estimate \(M_ 2(q;h)\leq qhP\) is sharp for \(1\leq h\leq \exp (cP)\) with a positive constant \(c\), the authors give the following theorem: For fixed \(k\geq 1\), \[ M_ k(q;h)\ll q(hP)^{k/2}-qhP. \] Exponential sums are used to derive a formula for \(M_ k(q;h)\), and the proof is then based on a fundamental lemma which gives an estimate for \(| \sum G_ 1(\rho_ 1)\dots G_ k(\rho_ k)|\) where \(G_ i(\rho_ i)\) are functions defined for \(\rho_ i=a/r_ i\) with \(1\leq a\leq r_ i\) and the summation is over all \(\rho_ i\) such that \(\rho_ i+\dots+\rho_ k\) is an integer. As the estimate is certainly of independent interest the authors promised a detailed discussion on the fundamental lemma elsewhere. For real \(\gamma\geq 1\), let \(V_{\gamma}(q)=\sum_{1\leq i\leq \varphi (q)}(a_{i+1}-a_ i)^{\gamma}\) where \(1\leq a_ i<a_ 2<..\). are integers coprime with q. In 1940 Erdős conjectured that \(V_ 2(q)\ll q^ 2/\varphi (q)\) and later \textit{C. Hooley} [Acta Arith. 8, 343--347 (1963; Zbl 0121.04706)] showed that \(V_{\gamma}(q)\asymp \varphi (q)P^{-\gamma}\) when \(1\leq \gamma <2\). As a corollary to their theorem the authors extend Hooley's result to any \(\gamma\geq 1\) which therefore includes the conjecture of Erdős.
scientific article; zbMATH DE number 3948401