Analysis of a Complex approximation to the Li-Keiper coefficients around the K Function

TL;DR

This paper introduces a perturbation approach for Li-Keiper coefficients around the K function, establishing a system of equations, verifying solutions, and supporting a stability conjecture through numerical evidence.

Contribution

It presents a novel perturbation method and a closed system of equations for Li-Keiper coefficients near the K function, with numerical validation of stability.

Findings

Established a closed system of equations for Li-Keiper coefficients.

Numerical evidence supports the stability conjecture with bounded fluctuations.

Verified solutions related to discrete derivatives of functions.

Abstract

We introduce a kind of "perturbation" for the Li-Keiper coefficients around the Koebe function (the K function) and establish a closed system of Equations for the Li-Keiper coefficients. We then check the correctness of some of the many possible solutions offered by the system ,related to the discrete derivative of order n of a function. We also report numerical finding which support our stability conjecture that the tiny part lambda-tiny(n) (the fluctuations around the trend) are bounded in absolute values by gammaxn, where gamma is the Euler-Mascheroni constant.