# Discretized Keiper/Li approach to the Riemann Hypothesis

## TL;DR

This paper introduces a discretized and explicit version of the Keiper--Li sequence, making it easier to analyze and compute, and provides tests that can detect zeros off the critical line related to the Riemann Hypothesis.

## Contribution

It develops a fully explicit, discretized version of the Keiper--Li sequence for better analysis and computation, and demonstrates its effectiveness in detecting zeros off the critical line.

## Key findings

- Sequences are fully explicit and computationally accessible up to n=500,000.
- The approach successfully detects zeros off the critical line in counterexamples.
- Provides explicit tests that react to non-Riemann zeros.

## 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.