A General Strong Nyman-beurling Criterion for the Riemann Hypothesis

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

Let ρ(x) = x − [x], χ = χ (0,1). In L2(0, ∞) consider the subspace B generated by {ρa|a ≥ 1} where ρa(x) := ρ 1 ax. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement χ ∈ B. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that χ ∈ B nat where B nat is the much smaller subspace...

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

A Divergent Vasyunin Correction

Abstract. V. I. Vasyunin has introduced special sequences of step functions related to the strong Nyman-Beurling criterion that converge pointwise to 1 in [1,∞). We show here that the first and simplest such sequence considered by Vasyunin diverges in L1((1,∞), x dx), which of course precludes the L2((1,∞), x dx)-convergence needed for the Riemann hypothesis. Whether all sequences considered by...

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 ...

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...