Von Mangoldt Function - Exponential Series

Exponential Series

Hardy and Littlewood examine the series

in the limit . Assuming the Riemann hypothesis, they demonstrate that

Curiously, they also show that this function is oscillatory as well, with diverging oscillations. In particular, there exists a value such that

and

infinitely often. The graphic to the right indicates that this behaviour is not at first numerically obvious: the oscillations are not clearly seen until the series is summed in excess of 100 million terms, and are only readily visible when .