A general strong Nyman-Beurling criterion for the Riemann hypothesis

@article{BezDuarte2005AGS,
  title={A general strong Nyman-Beurling criterion for the Riemann hypothesis},
  author={Luis B{\'a}ez-Duarte},
  journal={Publications De L'institut Mathematique},
  year={2005},
  volume={78},
  pages={117-125}
}
  • L. Báez-Duarte
  • Published 22 May 2005
  • Mathematics
  • Publications De L'institut Mathematique
For each [FORMULA] formally consider its Miintz transform [FORMULA]. For certain ƒ's with both [FORMULA] it is true that the Riemann hypothesis holds if and only if ƒ is in the L2 closure of the vector space generated by the dilations [FORMULA]. Such is the case for example when ƒ = X(0,1) where the above statement reduces to the strong Nyman criterion already established by the author. In this note we show that the necessity implication holds for any continuously differentiable function… 
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References

SHOWING 1-10 OF 21 REFERENCES
A CLOSURE PROBLEM RELATED TO THE RIEMANN ZETA-FUNCTION.
  • A. Beurling
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1955
TLDR
This note will deal with a duality of the indicated kind which may be of some interest due to its simplicity in statement and proof.
A Class of Invariant Unitary Operators
Abstract Let H = L 2 ((0, ∞),  dx ), and K λ f ( x )= f ( λx ), for λ >0, f ∈ H . An invariant operator on H is one commuting with all the K λ . A skew root is a self-adjoint, unitary operator on H
The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects
A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis
Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the
Étude de l’autocorrélation multiplicative de la fonction ‘partie fractionnaire’
A first encounter with $$A(\lambda):= \int_0^{+\infty} \{t\}\{\lambda t\} \frac{dt}{t^2}, \lambda > 0$$, a continuous function with a strict local maximum at every rational point.
(b)
On Fourier and Zeta(s)
Abstract We study some of the interactions between the Fourier Transform and the Riemann zeta function (and Dirichlet-Dedekind-Hecke-Tate L-functions).
A general statement of the functional equation for the Riemann zeta-function
A formal description of a functional analysis approach to the Riemann zeta-functional equation that provides in principle an infinity of different proofs based on work by the author on the existence
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