Von Mangoldt function

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

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

which thus gives the transcendental sequence

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

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

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

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

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

Dirichlet generating function

Since we have the Dirichlet series identity

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

the Dirichlet generating function of the von Mangoldt function is then

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

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

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

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

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

See also

• Summatory von Mangoldt function
• User:Peter Luschny/VonMangoldtTransformation