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.