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:
“If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.”
—Shoshana Zuboff (b. 1951)