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Wronskians, Cumulants, and the Riemann Hypothesis

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Constructive Approximation Aims and scope

Abstract

This paper proposes a new, analytic approach to the resolution of the Riemann hypothesis. The method has its origins in the 1986 results and techniques of Csordas, Norfolk, and Varga (CNV) in their proof of the Turán inequalities. Here, the mathematical structure of their work has been significantly extended and generalized. We make frequent use of the ideas on the sign-regularity of kernels found in the book of Karlin. The notion of cumulants plays an important role. The final step is to prove that a doubly infinite set of determinants are all positive.

We present a conjecture, supported by computations, about the sign-regularity of a set of cumulants of the function called Φ(t) by CNV. To illustrate the ideas, the conjecture is proved for an early member of the set. We describe a new method, superior to the Karlin method used by CNV, for proving positivity of the determinants, but some cases remain to be treated.

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Notes

  1. Unpublished reports by the author are available at http://publish.uwo.ca/~jnuttall.

References

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Acknowledgements

The assistance provided by the multiple precision arithmetic package written by David H. Bailey is much appreciated. The use of the Sharcnet consortium computing resources has been very helpful.

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  1. 3278 McGuffin Hills Dr., Parkhill, Ontario N0M 2K0, Canada

    John Nuttall

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  1. John Nuttall
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Correspondence to John Nuttall.

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Communicated by Edward B. Saff.

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Nuttall, J. Wronskians, Cumulants, and the Riemann Hypothesis. Constr Approx 38, 193–212 (2013). https://doi.org/10.1007/s00365-013-9200-8

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  • DOI: https://doi.org/10.1007/s00365-013-9200-8

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