Toward verification of the Riemann hypothesis: Application of the Li criterion

Abstract

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{λ_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η_j are valid. The constants η_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, not only does the Riemann hypothesis follow, but an inequality governing the values λ_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

to appear in Math. Physics, Analysis and Geometry; 1 figure

References in corpus (2)

• Effective method of computing Li's coefficients and their properties
• Polygamma theory, the Li/Keiper constants, and validity of the Riemann Hypothesis

Cited by in corpus (16)

• Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I
• Sharpenings of Li's criterion for the Riemann Hypothesis
• On Differences of Zeta Values
• Discretized Keiper/Li approach to the Riemann Hypothesis
• On certain sums over the nontrivial zeta zeros
• The difference between two Stieltjes constants
• A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants
• Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants
• The saddle-point method and the Li coefficients
• First applications of generalized Li's criterion to study the Riemann zeta-function zeroes location
• On harmonic binomial series
• An arithmetic interpretation of generalized Li's criterion
• A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the zeta constants
• Recurrence relations of Li coefficients
• Approximation of the derivatives of the logarithm of the Riemann zeta-function in the critical strip
• Generating functions for the generalized Li's sums