Discretized Keiper/Li approach to the Riemann Hypothesis

Abstract

The Keiper--Li sequence $\{{\lambda }_n\}$ is most sensitive to the Riemann Hypothesis asymptotically ($n\to \infty$), but highly elusive both analytically and numerically. We deform it to fully explicit sequences, simpler to analyze and to compute (up to $n=5\cdot 10^5$ by G. Misguich). This also works on the Davenport--Heilbronn counterexamples, thus we can demonstrate explicit tests that selectively react to zeros off the critical line.

39 pages, 11 figures. arXiv admin note: text overlap with arXiv:1602.03292. V2: corrections to definition of ${\mathcal{D}}_{R_0}$ (Sec.3.1), to uncertainty principle in Secs.1.2 (result unchanged) and 4.3; text revamped throughout. V3: introduction page added; Secs.3.3 & 4.3, figs.6 & 11 and bibliography augmented

References in corpus (6)

• Approximation of Riemann's zeta function by finite Dirichlet series: multiprecision numerical approach
• On Differences of Zeta Values
• Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis
• Simplifications of the Keiper/Li approach to the Riemann Hypothesis
• 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
• Asymptotic of the generalized Li's sums which non-negativity is equivalent to the Riemann Hypothesis

Cited by in corpus (1)

• Analyzing Riemann's hypothesis