Differential calculus for linear operators represented by finite signed measures and applications

We introduce a differential calculus for linear operators represented by a family of finite signed measures. Such a calculus is based on the notions of g-derived operators and processes and g-integrating measures, g?being a right-continuous nondecreasing function. Depending on the choice of?g, this differential calculus works for non-smooth functions and under weak integrability conditions. For linear operators represented by stochastic processes, we provide a characterization criterion of g-differentiability in terms of characteristic functions of the random variables involved. Various illustrative examples are considered. As an application, we obtain an efficient algorithm to compute the Riemann zeta function ??(z) with a geometric rate of convergence which improves exponentially as ?(z) increases.