A Strengthening of the Nyman-beurling Criterion for the Riemann Hypothesis

A Strengthening of the Nyman-beurling Criterion for the Riemann

. By the NymanBeurling criterion the Riemann hypothesis is equivalent to the statement χ ∈ B. For some time it has been conjectured, and proved in the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that χ ∈ Bnat where B is the much smaller subspace generated by {ρa|a ∈ N}. This second version differs from the ...

Institute for Mathematical Physics the Nyman{beurling Equivalent Form for the Riemann Hypothesis the Nyman-beurling Equivalent Form for the Riemann Hypothesis

A General Strong Nyman-beurling Criterion for the Riemann Hypothesis

For each f : [0,∞) → C formally consider its co-Poisson or Müntz transform g(x) = ∑ n≥1 f(nx) − 1 x ∫∞ 0 f(t)dt. For certain f ’s with both f, g ∈ L2(0,∞) it is true that the Riemann hypothesis holds if and only if f is in the L2 closure of the vector space generated by the dilations g(kx), k ∈ N. Such is the case for example when f = χ(0,1] where the above statement reduces to the strong Nyman...

On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis

Abstract. There has been a surge of interest of late in an old result of Nyman and Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many authors have contributed to this circle of ideas, culminating in a beautiful refinement due to Baez-Duarte. The purpose of this little survey is to dis-entangle the resulting web of complications, and reveal the essential simplicity of th...

On an analytic estimate in the theory of the Riemann zeta function and a theorem of Báez-Duarte.

On the Riemann hypothesis we establish a uniform upper estimate for zeta(s)/zeta (s + A), 0 < or = A, on the critical line. We use this to give a purely complex-analytic variant of Báez-Duarte's proof of a strengthened Nyman-Beurling criterion for the validity of the Riemann Hypothesis. We investigate function-theoretically some of the functions defined by Báez-Duarte in his study and we show t...