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Mangoldt function

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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://www.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