Beurling’s theorem, Davenport’s formula, and the Riemann hypothesis

Abstract:

Various unitarily equivalent models are treated that are related to the approach of Beurling and Nyman to the Riemann hypothesis about zeros of the Riemann zeta function. The relationship is discussed between the Riemann hypothesis and properties of the subspace of the weighted Hilbert space $L^2_{1/x^2}(0, 1)$ generated by the functions $\rho (nx)$, $n=1, 2, \dots$, where $\rho ( \cdot )$ denotes the fractional part of a real number.