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# Von Mangoldt function

The von Mangoldt function ${\displaystyle \scriptstyle M(n)\,}$, also called the lambda function ${\displaystyle \scriptstyle \Lambda (n)\,}$, is the function of positive integers ${\displaystyle \scriptstyle n\,}$ defined by

${\displaystyle \Lambda (n)={\begin{cases}\log(p)&{\mbox{if }}n=p^{k}{\mbox{ for some prime }}p{\mbox{ and integer }}k\geq 1,\\0&{\mbox{otherwise.}}\end{cases}}}$

which thus gives the transcendental sequence

{${\displaystyle \scriptstyle 0,\log(2),\log(3),\log(2),\log(5),0,\log(7),\log(2),\log(3),0,\log(11),0,\log(13),0,0,\log(2),\log(17),0,\log(19),0,0,0,\log(23),0,\log(5),0,\log(3),0,\log(29),0,...\,}$}

## Exponential of the von Mangoldt function

A014963 Exponential of the von Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power when a(n) = that prime.

{1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, ...}

which is given by the formula

${\displaystyle e^{\Lambda (n)}={\frac {LCM(1,2,...,n)}{LCM(1,2,...,n-1)}},\,}$

where ${\displaystyle \scriptstyle LCM(1,2,...,n)\,}$ is the least common multiple function.

The GCD of all the interior cells of the ${\displaystyle \scriptstyle n\,}$th row of Pascal's triangle gives the exponential of the von Mangoldt function. ( (Verify: THIS CONJECTURE NEEDS TO BE CONFIRMED....)[1])

## Dirichlet generating function

Since we have the Dirichlet series identity

${\displaystyle {{\bigg [}\log {\bigg (}{\frac {1}{\zeta (s)}}{\bigg )}}{\bigg ]}^{'}=-{{\big [}\log {\big (}\zeta (s){\big )}}{\big ]}^{'}=-{\frac {\zeta ^{'}(s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}},\ \Re (s)>1,\,}$

the Dirichlet generating function of the von Mangoldt function is then

${\displaystyle D_{\{\Lambda (n)\}}(s)={{\bigg [}\log {\bigg (}{\frac {1}{\zeta (s)}}{\bigg )}}{\bigg ]}^{'}=-{{\big [}\log {\big (}\zeta (s){\big )}}{\big ]}^{'}=-{\frac {\zeta ^{'}(s)}{\zeta (s)}}.\,}$

Also, since the Dirichlet generating function of the Möbius function is

${\displaystyle D_{\{\mu (n)\}}(s)={\frac {1}{\zeta (s)}},\,}$

we thus have the following relation between the Dirichlet generating function of the von Mangoldt function and the Dirichlet generating function of the Möbius function

${\displaystyle D_{\{\Lambda (n)\}}(s)={{\big [}\log {\big (}D_{\{\mu (n)\}}(s){\big )}}{\big ]}^{'}\,}$