For efficiency, we compute

where

is the Euler totient function [5], and precompute the inside sum [7].

In all,

values of the Mertens function

are plotted as red bars in intervals of 100 per visual frame [6].

Corresponding values of the distribution measure

are overlaid in yellow to compare the two function values.

Where one colored bar hides the other fully, use the difference

plotted below as green bars to compare values.

For

, the heights of the red and yellow bars show correlations and patterns as expected from mappings that exist between the Möbius function

and subsets of the Farey sequence

with denominators

[4].

For

,

[4], suggesting observable patterns continuing to this range.

In [4], it is seen from Ramanujan's sums that

, where

runs over the set

.

Let

; then

binds the two functions.

The distribution of the primes gives the distribution of the Farey sequence via the prime reciprocals.

The distribution of the Möbius function and its summation, the Mertens function, have mysterious random-like properties.

[7] L. Quet, "Sum of Positive Integers, k, where k <= n/2 and GCD(k,n)=1." (Jan 20, 2002)

oeis.org/A066840.