Bayes meet Riemann- Bayesian Characterization of Infinite Series with Application to Riemann Hypothesis

TL;DR

This paper introduces a novel Bayesian approach to determine the convergence or divergence of infinite series, including oscillating series, and applies it to the Riemann Hypothesis, providing new insights and partial support for the conjecture.

Contribution

It develops a Bayesian framework for assessing infinite series convergence, capable of handling cases where traditional tests fail, and extends this to analyze the Riemann Hypothesis.

Findings

Bayesian method successfully assesses convergence in difficult cases.

Application to Riemann Hypothesis yields results that do not fully support the conjecture.

Extended to oscillating series with multiple limit points.

Abstract

In the classical literature on infinite series there are various tests to determine if a given infinite series converges, diverges, or oscillates. But unfortunately, for very many infinite series all the existing tests can fail to provide definitive answers. In this article we propose a novel Bayesian theory for assessment of convergence properties of any given infinite series. Remarkably, this theory attempts to provide conclusive answers to the question of convergence even where all the existing tests of convergence fail. We apply our ideas to seven different examples, obtaining very encouraging results. Importantly, we also apply our ideas to investigate the Riemann Hypothesis, and obtain results that do not completely support the conjecture. We also extend our ideas to develop a Bayesian theory on oscillating series, where we allow even infinite number of limit points. Analysis of Riemann Hypothesis using Bayesian multiple limit points theory yielded almost identical results as the Bayesian theory of convergence assessment.