First applications of generalized Li's criterion to study the Riemann\n zeta-function zeroes location

2014/04/29 by S. K. Sekatskiǐ, Sekatskii, Sergey · 2 citations
Mathematics · #Algebraic and Geometric Analysis #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1404.7276

openalex publication_date 2014/04/29 · openalex created_date 2022/09/05 · openalex updated_date 2026/07/28

Abstract

We present the first applications of the recently established by us\n(arXiv:1304.7895; Ukrainian Math. J. - 2014. -66. - P. 371-383) generalized\nLi's criterion equivalent to the Riemann Hypothesis. This criterion is the\nstatement that the Riemann hypothesis is equivalent to the non-negativity of\nthe derivatives 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln(\ξ(z))) for z=b+1 of the\nRiemann xi-function for all real b>-1/2 and all m = 1, 2, 3... We show that for\nany positive integer n there is such value of b_n (depending on n) that for all\nm<=n and b>b_n, inequality 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln(\ξ(z))) for\nz=b+1 >=0 does hold true. Assuming RH, we also have found an asymptotic of the\ngeneralized Li's sums over non-trivial Rieman zeroes for large n, and discuss\nwhat asymptotic of 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln((z-1)*(\ζ(z))) at the\npoint z+b+1 is required for the Riemann hypothesis holds true.\n

Citations

• The Positivity of a Sequence of Numbers and the Riemann Hypothesis
• Tables of Integrals, Series, and Products

Cited by

• Analysis of Voros criterion: what derivatives involving the logarithm of the Riemann xi-function at z=1/2 should be non-negative for the Riemann hypothesis holds true
• Approximation of the derivatives of the logarithm of the Riemann zeta-function in the critical strip

Related

• Generalized Bombieri - Lagarias' theorem and generalized Li's criterion
• Approximation of the derivatives of the logarithm of the Riemann zeta-function in the critical strip
• An arithmetic interpretation of generalized Li's criterion
• Asymptotic of the generalized Li's sums which non-negativity is equivalent to the Riemann Hypothesis
• A sharpening of Li's criterion for the Riemann Hypothesis