An Integral Equation for Riemann's Zeta Function and Its Approximate Solution. (15th May 2020)

• Record Type: Journal Article
• Authors: Milgram, Michael
• Other Names: Hu Ying Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2020(2020)
• Publication Date: 2020-05-15
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2020/1832982
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : 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 + i t, 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 + i t for different values of σ and equal values of t ; this is illustrated in a number of figures.