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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
- ↑ Needs verification (THIS CONJECTURE NEEDS TO BE CONFIRMED...).