View source for Mangoldt function

The [[Arithmetic function|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:
$$
\sum_{d | n} \Lambda(d) = \log n
$$
$$
\Lambda(n) = \sum_{d|n} \mu(d) \log(n/d)
$$
where the sums are taken over all divisors $d$ of $n$. The Mangoldt function is closely connected with the Riemann [[Zeta-function|zeta-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.



====Comments====
In the article above, $\mu$ denotes the [[Möbius function|Möbius function]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Sect. 17.7</TD></TR>
</table>