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International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 7, Pages 387-406

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

Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado 21827, Caracas 1020-A, Venezuela

Received 15 May 2001; Revised 19 February 2002

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let ρ(x)=x[x], χ=χ(0,1), λ(x)=χ(x)logx, and M(x)=ΣKxμ(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 s>1/p is equivalent to M1r< for all r(1,p). The space is the linear space generated by the functions xρ(θ/x) with θ(0,1]. Define Gn(x)=1/n1M1(θ)ρ(θ/x)θ1dθ. For all p(1,) we prove the following theorems: (I) M1p< implies λ¯Lp, and λ¯Lp implies M1r< for all r(1,p). (II) Gnλp0 implies ξ(s)0 in s1/p, and ξ(s)0 in s1/p implies Gnλr0 for all r(1,p).