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

Abstract

Let ρ(x)=x−[x], χ=χ(0,1), λ(x)=χ(x)logx, and M(x)=ΣK≤x μ(k), where μ is the Möbius function. Norms are in Lp(0,∞), 1<p<∞. For M1(θ)=M(1/θ) it is noted that ξ(s)≠0 in ℜs>1/p is equivalent to ‖M1‖r<∞ for all r∈(1,p). The space ℬ is the linear space generated by the functions x↦ρ(θ/x) with θ∈(0,1]. Define Gn(x)=∫1/n1M1(θ)ρ(θ/x)θ−1 dθ. For all p∈(1,∞) we prove the following theorems: (I) ‖M1‖p<∞ implies λ∈ℬ¯Lp, and λ∈ℬ¯Lp implies ‖M1‖r<∞ for all r∈(1,p). (II) ‖Gn−λ‖p→0 implies ξ(s)≠0 in ℜs≥1/p, and ξ(s)≠0 in ℜs≥1/p implies ‖Gn−λ‖r→0 for all r∈(1,p).