What is it about?
The criterion put forward independently by Keiper and Li as to the truth of the Riemann hypothesis provides a way of combining rigorous analysis with sophisticated numerics. A new way of looking at this criterion splits the calculation of an infinite set of coefficients due to Li into the product of two parts: one combinatoric in nature, the other consisting of integer powers multiplying well-studied constants. We show in this paper how the second part can be dealt with numerically to high accuracy, and asymptotically to high order.
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Why is it important?
The Riemann hypothesis is widely regarded as the most important unsolved problem in mathematics, with its ramifications spilling over into kindred subjects. The Keiper-Li criterion is a relatively new and attractive way of studying this problem: attractive because its combination of analytics and numerics permits modern tools to be brought to bear. We show here that a substantial step forward can be made using a classical tool (Lambert functions), recently extended in a way which has already yielded valuable results. The work reported here augurs well for the results from generalized Lambert functions in relation to the Keiper-Li criterion.
Perspectives
The numerical investigations in the literature of the Keiper-Li criterion show its potential as a way to better understand the Riemann hypothesis. The results reported here emphasize and clarify that potential in a very satisfactory way.
Dr Tony Cyril Scott
RWTH-Aachen University
The Keiper-Li criterion is an important way of investigating the Riemann hypothesis. The results presented using generalised Lambert functions enable high accuracy investigations of the elements of this criterion to be evaluated quickly. They should enable further research into the criterion to be carried out more easily, and hopefully decisively.
Ross McPhedran
University of Sydney
Read the Original
This page is a summary of: The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions, ACM Communications in Computer Algebra, September 2023, ACM (Association for Computing Machinery),
DOI: 10.1145/3637529.3637530.
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