Difference between revisions of "Mangoldt function"
From Encyclopedia of Mathematics
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The [[Arithmetic function|arithmetic function]] defined by | The [[Arithmetic function|arithmetic function]] defined by | ||
| − | + | $$ | |
| − | + | \Lambda(n) = \begin{cases} \log p &\mbox{if } n = p^m,\,p \mbox{ prime},\,m\ge 1 \\ | |
| − | + | 0 & \mbox{otherwise} . \end{cases} | |
| − | The function | + | $$ |
| − | + | 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]] | |
| − | where the sums are taken over all divisors | + | $$ |
| − | + | \Lambda(n) = \sum_{d|n} \mu(d) \log(n/d) | |
| − | + | $$ | |
| − | + | 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. | The Mangoldt function was proposed by H. Mangoldt in 1894. | ||
| + | ====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> | ||
| − | + | {{TEX|done}} | |
| − | |||
| − | + | [[Category:Number theory]] | |
| − | |||
Latest revision as of 18:33, 18 October 2014
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=14110
Mangoldt function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mangoldt_function&oldid=14110
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article