Algosim documentation: Mertens

Mertens

The Mertens function.

Syntax

Description

If n is a positive integer, then Mertens(n) is defined as the number of square-free positive integers up to n with an even number of prime factors minus the number of square-free positive integers up to n with an odd number of prime factors.

Hence,

Mertens(n) = ∑(MöbiusMu(k), k, 1, n).

Examples

Mertens(100000)
−48
SequenceVector(20) @ Mertens
(1, 0, −1, −1, −2, −1, −2, −2, −2, −1, −2, −2, −3, −2, −1, −1, −2, −2, −3, −3)
count(SequenceVector(1000) @ Mertens, IsPositive)
254
collapse(SequenceVector(1000) @ Mertens @ sgn)
(1, 1)
(0, 1)
(−1, 36)
(0, 2)
(−1, 17)
(0, 1)
(−1, 6)
(0, 1)
(−1, 27)
(0, 1)
(1, 7)
⋮

See also