On some positivity properties of the Riemann Ξ function
Abstract
We prove strict log-concavity of the positive functions Θ and Λ with one-sided and bilateral Laplace transforms 1/ξ(1/2+√s) and 1/Ξ, respectively, where Ξ(z)=ξ(1/2+iz). For either function, log-concavity is the Pólya frequency property of order two (PF2). Without assuming the Riemann hypothesis (RH), but using its published verification up to height about 3·1012, we further prove that both densities are PFk* for k*=1 810 540 931 075. The proof views Θ as a small signed perturbation of a density that is PF of every order, with an error bound independent of the order, so the numerical check reduces to scalar inequalities. Since Λ is a Gaussian mixture of Θ, it inherits the same order. Conversely, a pole obstruction shows that if either density is PFk, then every zeta zero lower than the (k+2)-nd lies on the critical line. Finally, we prove that if RH holds up to height T>1/2, then Ξ satisfies the generalized Laguerre inequalities Ln[Ξ](x)≥0 of every order n for |x|≤ T−1/2, extending the case n=1 of Csordas, Ruttan and Varga (1991).
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BibTeX
@misc{sra2026some,
title = {{On some positivity properties of the Riemann $Ξ$ function}},
author = {Suvrit Sra},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02944},
eprint = {2611.02944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02944},
}
RIS
TY - UNPB
TI - On some positivity properties of the Riemann Ξ function
AU - Suvrit Sra
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02944
UR - https://arxiv.org/abs/2611.02944
AB - We prove strict log-concavity of the positive functions Θ and Λ with one-sided and bilateral Laplace transforms 1/ξ(1/2+√s) and 1/Ξ, respectively, where Ξ(z)=ξ(1/2+iz). For either function, log-concavity is the Pólya frequency property of order two (PF_2). Without assuming the Riemann hypothesis (RH), but using its published verification up to height about 3·10^12, we further prove that both densities are PF_k_* for k_*=1 810 540 931 075. The proof views Θ as a small signed perturbation of a density that is PF of every order, with an error bound independent of the order, so the numerical check reduces to scalar inequalities. Since Λ is a Gaussian mixture of Θ, it inherits the same order. Conversely, a pole obstruction shows that if either density is PF_k, then every zeta zero lower than the (k+2)-nd lies on the critical line. Finally, we prove that if RH holds up to height T>1/2, then Ξ satisfies the generalized Laguerre inequalities L_n[Ξ](x)≥0 of every order n for |x|≤ T−1/2, extending the case n=1 of Csordas, Ruttan and Varga (1991).
ER -