Moebius Function
The classical Moebius and Mertens functions in number theory.
moebius(n) mertens(n)
n |
Positive integer. |
moebius(n) is +1 if n is a square-free positive integer
with an even number of prime factors, or +1 if there are an odd
of prime factors. It is 0 if n is not square-free.
mertens(n) is the aggregating summary function, that sums up all
values of moebius from 1 to n.
For moebius, 0, 1 or -1, depending on the prime
decomposition of n.
For mertens the values will very slowly grow.
Works well up to 10^9, but will become very slow for the Mertens
function.
sapply(1:16, moebius) sapply(1:16, mertens) ## Not run: x <- 1:50; y <- sapply(x, moebius) plot(c(1, 50), c(-3, 3), type="n") grid() points(1:50, y, pch=18, col="blue") x <- 1:100; y <- sapply(x, mertens) plot(c(1, 100), c(-5, 3), type="n") grid() lines(1:100, y, col="red", type="s") ## End(Not run)
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