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

Let ρ ( x ) = x − [ x ] , χ = χ ( 0 , 1 ) , λ ( x ) = χ ( x ) log x , and M ( x ) = Σ K ≤ x μ ( k ) , where μ is the Möbius function. Norms are in L p ( 0 , ∞ ) , 1 < p < ∞ . For M 1 ( θ ) = M ( 1 / θ ) it is noted that ξ ( s ) ≠ 0 in ℜ s > 1 / p is equivalent to ‖ M 1 ‖ 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 G n ( x ) = ∫ 1 / n 1 M 1 ( θ ) ρ ( θ / x ) θ − 1 d θ . For all p ∈ ( 1 , ∞ ) we prove the following theorems: (I) ‖ M 1 ‖ p < ∞ implies λ ∈ ℬ ¯ L p , and λ ∈ ℬ ¯ L p implies ‖ M 1 ‖ r < ∞ for all r ∈ ( 1 , p ) . (II) ‖ G n − λ ‖ p → 0 implies ξ ( s ) ≠ 0 in ℜ s ≥ 1 / p , and ξ ( s ) ≠ 0 in ℜ s ≥ 1 / p implies ‖ G n − λ ‖ r → 0 for all r ∈ ( 1 , p ) .