Namespaces
Variants
Actions

Mangoldt function

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

The arithmetic function defined by $$ \Lambda(n) = \begin{cases} \log p &\mbox{if } n = p^m,\,p \mbox{ prime},\,m\ge 1 \\ 0 & \mbox{otherwise} . \end{cases} $$ The function $\Lambda(n)$ has the following properties: $$ \sum_{d | n} \Lambda(d) = \log n \,, $$ where $\mu$ denotes the Möbius function, and so by Möbius inversion $$ \Lambda(n) = \sum_{d|n} \mu(d) \log(n/d) $$ where the sums are taken over all divisors $d$ of $n$. The Mangoldt function is closely connected with the Riemann zeta-function $\zeta(s)$. In fact, the generating series for $\Lambda(n)$ is the logarithmic derivative of $\zeta(s)$: $$ -\frac{\zeta'(s)}{\zeta(s)} = \sum_n \Lambda(n) n^{-s}\ \ \ (\Re s > 1) $$ The Mangoldt function was proposed by H. Mangoldt in 1894.


References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Sect. 17.7
How to Cite This Entry:
Mangoldt function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mangoldt_function&oldid=33833
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article