An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution

• Main Author: Michael Milgram
• Format: Article
• Language: English
• Published: Hindawi Limited 2020-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://dx.doi.org/10.1155/2020/1832982

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+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 ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures.