Abstract view
On the size of an expression in the NymanBeurlingBáezDuarte criterion for the Riemann Hypothesis


Published:20180111
Printed: Sep 2018
Helmut Maier,
Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany
Michael Th. Rassias,
Institute of Mathematics, University of Zurich, CH8057, Zurich, Switzerland
Abstract
A crucial role in the NymanBeurlingBáezDuarte approach to
the Riemann Hypothesis is played by the distance
\[
d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{\infty}^\infty
\left1\zeta A_N
\left(\frac{1}{2}+it
\right)
\right^2\frac{dt}{\frac{1}{4}+t^2}\:,
\]
where the infimum is over all Dirichlet polynomials
$$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$
of length $N$.
In this paper we investigate $d_N^2$ under the assumption that
the Riemann zeta function has four nontrivial zeros off the
critical line.
MSC Classifications: 
30C15, 11M26 show english descriptions
Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
30C15  Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 11M26  Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
