|
|||||
|
|
Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis |
|
| Institution: | 1. Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany;2. Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland;1. Research Center for Operator Algebras and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Department of Mathematics, East China Normal University, 3663 Zhongshan North Road, Putuo District, Shanghai 200062, China;2. Department of Mathematics & Statistics, University of Windsor, Windsor, ON N9B 3P4, Canada;1. Lomonosov Moscow State University, Russian Federation;2. National Research University Higher School of Economics, Russian Federation;3. Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany |
| Abstract: | We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series. |
| Keywords: | Riemann Hypothesis Riemann zeta function Nyman–Beurling–Báez–Duarte criterion |
| 本文献已被 ScienceDirect 等数据库收录！ | |