New versions of the Nyman-Beurling criterion for the Riemann hypothesis
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...
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202013248 |
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| _version_ | 1832565678869053440 |
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| author | Luis Báez-Duarte |
| author_facet | Luis Báez-Duarte |
| author_sort | Luis Báez-Duarte |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-d4e45f21844c4deaa70b3a032c932eda |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d4e45f21844c4deaa70b3a032c932eda2025-02-03T01:07:04ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0131738740610.1155/S0161171202013248New versions of the Nyman-Beurling criterion for the Riemann hypothesisLuis Báez-Duarte0Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado 21827, Caracas 1020-A, VenezuelaLet 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).http://dx.doi.org/10.1155/S0161171202013248 |
| spellingShingle | Luis Báez-Duarte New versions of the Nyman-Beurling criterion for the Riemann hypothesis International Journal of Mathematics and Mathematical Sciences |
| title | New versions of the Nyman-Beurling criterion for the Riemann hypothesis |
| title_full | New versions of the Nyman-Beurling criterion for the Riemann hypothesis |
| title_fullStr | New versions of the Nyman-Beurling criterion for the Riemann hypothesis |
| title_full_unstemmed | New versions of the Nyman-Beurling criterion for the Riemann hypothesis |
| title_short | New versions of the Nyman-Beurling criterion for the Riemann hypothesis |
| title_sort | new versions of the nyman beurling criterion for the riemann hypothesis |
| url | http://dx.doi.org/10.1155/S0161171202013248 |
| work_keys_str_mv | AT luisbaezduarte newversionsofthenymanbeurlingcriterionfortheriemannhypothesis |