Period-harmonic-tupling jumps to chaos and fractal-scaling in a class of series

Series like Riemann zeta function are found to have large jumps at harmonic periods. The route to chaos for these series is via the cascade of period-harmonic-tupling jumps in contrast to the cascade of period-doubling bifurcations for unimodal iterative maps like the logistic map. The sets of oscillations between one jump and the next also exhibits the fractal-scaling property. This gives rise to the simplest of all approximations of Riemann zeta function where .