Extending a Function Just by Multiplying and Dividing Function Values: Smoothness and Prime Identities

Abstract

We describe a purely-multiplicative method for extending an analytic function. It calculates the value of an analytic function at a point, merely by multiplying together function values and reciprocals of function values at other points closer to the origin. The function values are taken at the points of geometric sequences, independent of the function, whose geometric ratios are arbitrary. The method exposes an "elastic invariance" property of all analytic functions. We show how to simplify and truncate multiplicative function extensions for practical calculations. If we choose each geometric ratio to be the reciprocal of a power of a prime number, we obtain a prime functional identity, which contains a generalization of the M\"obius function (with the same denominator as the Rieman zeta function), and generates prime number identities.