2020 An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution
Michael Milgram
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Abstr. Appl. Anal. 2020: 1-29 (2020). DOI: 10.1155/2020/1832982


Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξ(s) function and thus ζ(s) indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζ(s) anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζ(σ+it), everywhere inside the asymptotic (t) critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of ζ(σ+it) for different values of σ and equal values of t; this is illustrated in a number of figures.


I am grateful to Larry Glasser, who early on provided me with a bespoke derivation of (20); shortly afterwards, he outdid himself by pointing out Romik’s paper [13], thereby negating his invitation to pen a guest Appendix. I also thank Larry, who identified the value of the Mellin transforms, and Vini Anghel for commenting on a preliminary version of this manuscript. All expenses associated with the work presented here have been borne by myself. I am therefore grateful to the Board of Director of Geometrics Unlimited, Ltd. who have agreed to finance any further (i.e., publication) costs.


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Michael Milgram. "An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution." Abstr. Appl. Anal. 2020 1 - 29, 2020. https://doi.org/10.1155/2020/1832982


Received: 23 October 2019; Accepted: 6 January 2020; Published: 2020
First available in Project Euclid: 28 July 2020

Digital Object Identifier: 10.1155/2020/1832982

Rights: Copyright © 2020 Hindawi


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