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Von Mangoldt function

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The von Mangoldt function , also called the lambda function , is the function of positive integers defined by

which thus gives the transcendental sequence

{}

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

where is the least common multiple function.

The GCD of all the interior cells of the 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

the Dirichlet generating function of the von Mangoldt function is then

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

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

See also

Notes

  1. Needs verification (THIS CONJECTURE NEEDS TO BE CONFIRMED...).