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
Famous quotes containing the word series:
“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)