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Bayes Meets Riemann – Bayesian Characterization of Infinite Series with Application to Riemann Hypothesis

Sucharita Roy, Sourabh Bhattacharya


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.


Bayesian theory; Dirichlet process; Infinite series; M¨obius function; Riemann Hypothesis; Tests of series convergence.

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