New versions of the Nyman-Beurling criterion for the Riemann hypothesis

Abstract

Letρ(x)=x−[x],χ=χ(0,1),λ(x)=χ(x)logx, andM(x)=ΣK≤x μ(k), whereμis the Möbius function. Norms are inLp(0,∞),1<p<∞. ForM1(θ)=M(1/θ)it is noted thatξ(s)≠0inℜs>1/pis equivalent to‖M1‖r<∞for allr∈(1,p). The spaceℬis the linear space generated by the functionsx↦ρ(θ/x)withθ∈(0,1]. DefineGn(x)=∫1/n1M1(θ)ρ(θ/x)θ−1 dθ. For allp∈(1,∞)we prove the following theorems: (I)‖M1‖p<∞impliesλ∈ℬ¯Lp, andλ∈ℬ¯Lpimplies‖M1‖r<∞for allr∈(1,p). (II)‖Gn−λ‖p→0impliesξ(s)≠0inℜs≥1/p, andξ(s)≠0inℜs≥1/pimplies‖Gn−λ‖r→0for allr∈(1,p).