A numerical comment on the 'tiny' oscillations and a heuristic 'conjecture'

TL;DR

This paper analyzes the tiny oscillations of Li-Keiper coefficients, adds new computed values, and proposes a stronger numerical conjecture with logarithmic behavior, supported by preliminary numerical experiments.

Contribution

It introduces a more robust numerical conjecture on the oscillations of Li-Keiper coefficients and provides additional computed data to support this hypothesis.

Findings

New values for lambda5 and lambda6 coefficients are presented.

A stronger numerical conjecture with logarithmic behavior is proposed.

Preliminary numerical experiments support the conjecture.

Abstract

In the first part of this short work (in the form of a comment) we add the plots of two more values of the Li-Keiper coefficients lambda5 and lambda6, computed as in our recent work where the first four values were in particular given. This for the trend as well as for the oscillating path ('tiny', the term coined by Maslanka in his pioneering work). Then, in the second part looking at the tiny oscillations, we propose a 'numerical conjecture' in a more strong form, i.e. with a logarithmic behaviour and carry out a short numerical experiment on the new 'numerical conjecture'.