Dirichlet Series
The von Mangoldt function plays an important role in the theory of Dirichlet series, and in particular, the Riemann zeta function. In particular, one has
for . The logarithmic derivative is then
These are special cases of a more general relation on Dirichlet series. If one has
for a completely multiplicative function, and the series converges for, then
converges for .
Read more about this topic: Von Mangoldt Function
Famous quotes containing the word series:
“A sophistical rhetorician, inebriated with the exuberance of his own verbosity, and gifted with an egotistical imagination that can at all times command an interminable and inconsistent series of arguments to malign an opponent and to glorify himself.”
—Benjamin Disraeli (1804–1881)