Curve Fitting, Differential Equations And The Riemann Hypothesis

Abstract

It is known that the Riemann zeta function ζ (s) in the critical strip 0 < Re(s) < 1, may be represented as the Mellin transform of a certain function φ (x) which is related to one of the theta functions. The function φ (x) satisfies a well known functional equation, and guided by this property we deduce a family of approximating functions involving an arbitrary parameter α. The approximating function corresponding to the value of α = 2 gives rise to a particularly accurate numerical approximation to the function φ (x). Another approximation to φ (x), which is based upon the first one, is obtained by solving a certain differential equation. Yet another approximating function may be determined as a simple extension of the first. All three approximations, when used in conjunction with the Mellin transform expression for ζ (s) in the critical strip, give rise to an explicit expression from which it is clear that Re(s) = 1/2 is a necessary and sufficient condition for the vanishing of the imaginary part of the integral, the real part of which is non-zero. Accordingly, the analogy with the Riemann hypothesis is only partial, but nevertheless Re(s) = 1/2 emerges from the analysis in a fairly explicit manner. While it is generally known that the imaginary part of the Mellin transform must vanish along Re(s) = 1/2, the major contribution of this paper is the presentation of the actual calculation for three functions which approximate φ (x). The explicit nature of these calculation details may facilitate progress towards the corresponding calculation for the actual φ (x), which may be necessary in a resolution of the Riemann hypothesis.