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von Mangoldt’s function

The von Mangoldt function  is defined by

Λ(n) = { k log p, if n = p , 0, otherwise(1)

where typeset structure stands for a prime and typeset structurean integer.

The Moebius inversion formula gives

Λ(n) = Underoverscript[∑, d | n, arg3] μ(n/d) log d = -Underoverscript[∑, d | n, arg3] μ(d) log d .(2)

Here typeset structure stands for the Moebius function.

If typeset structure we have

Underoverscript[∑, d | n, arg3] Λ(d) = log n .(3)

The average order of typeset structure is 1, that is

Underscript[lim, x -> ∞] 1/x Underoverscript[∑, n <= x, arg3] Λ(n) = 1 .(4)

We also have

Underoverscript[∑, n <= x, arg3] (Λ(n) log n)/n = 1/2 log^2 x + O(log x) .(5)

and for typeset structure

Underoverscript[∑, n <= x, arg3] Λ(n) ⌊ x/n ⌋ = log ⌊ x ⌋ != x log x - x + O(log x) .(6)

Cite this web-page as:

Štefan Porubský: von Mangoldt’s Function.

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