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 .
Read more about this topic: Von Mangoldt Function
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