Mangoldt function

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1: 27.2 Functions

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27.2.14 $$\mathrm{\Lambda }\left(n\right)=\ln p,$$ $n=p^a$,

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Defines:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E14

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►This is Mangoldt’s function. …

2: 27.5 Inversion Formulas

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27.5.5 $$\ln n=\sum _{d|n}\mathrm{\Lambda }\left(d\right)⟺\mathrm{\Lambda }\left(n\right)=\sum _{d|n}(\ln d)\mu \left(\frac{n}{d}\right).$$

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Symbols:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function, $\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.5.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.5 and Ch.27

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3: 27.4 Euler Products and Dirichlet Series

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27.4.12 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{\Lambda }\left(n\right)n^{-s}=-\frac{{\zeta }^{\prime }\left(s\right)}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Referenced by:

§27.4

Permalink:

http://dlmf.nist.gov/27.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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4: 27.3 Multiplicative Properties

… ►Except for $\nu \left(n\right)$, $\mathrm{\Lambda }\left(n\right)$, $p_n$, and $\pi \left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and …