Explicit estimates of sums related to the Nyman-Beurling criterion for the Riemann hypothesis (Q1740621)

The main result of the paper is the following. Theorem. Let \(D\geq 2\) and \(C\) be the number which is uniquely determined by \[ C\geq \frac{\sqrt{5}+1}{2}, \quad 2C-\mbox{log}\; 2=\frac{1}{2} \log 2. \] Let \(v_0\) be determined by \[ v_0 \left( 1-\left( 1+2\mbox{log}\; 2 \left( C+\frac{\log 2}{2} \right)^{-1}\right)^{-1}+2+\frac{4}{\log 2}C\right)=2. \] Let \(z_0= 2-\left( 2+ \frac{4}{\log\ 2}C \right) v_0.\) Then for all \(\varepsilon>0\) we have \[ \sum\limits_{k^D \leq n<2k^D } \mu(n) g\left(\frac{n}{k} \right)\ll_{\varepsilon }k^{D-z_0+\varepsilon}.\tag{1} \] Here \(\mu\) is the Möbius function and \[ g(z) = \sum\limits_{l\geq 1} \frac{1-2\{lz\}}{ l}. \] The sum on the left hand side of (1) appears in the Nyman-Beurling criterion for the Riemann hypothesis [\textit{L. Báez-Duarte} et al., Adv. Math. 149, No. 1, 130--144 (2000; Zbl 1008.11032)]. The estimate (1) is remarkably sharp in comparison to other sums containing the Möbius function. In the proof the theory of continued fractions and the theory of Fourier series are intensively used.