Correlating the Mertens Function with the Farey Sequence
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The Möbius function is defined to be if is the product of distinct primes, and zero otherwise, with . The values of the Möbius function are for positive integers .[more]
The Mertens function is defined to be the cumulative sum of the Möbius function, , so that the values of the Mertens function are for positive integers . (In this Demonstration we start the sequence at .)
The Farey sequence of order is the set of irreducible fractions between 0 and 1 with denominators less than or equal to , arranged in increasing order . For , the new terms are , , , , , . Therefore, .
Truncating the Farey sequence to include only the fractions less than and omitting 0 and 1, define . Define a measure of the Farey sequence distribution by that describes an asymmetry in the distribution of about . Then the values of are .
This Demonstration compares the Mertens function values (red) with (yellow), and shows the difference in green. The values shown range from to .[less]
Contributed by: Jenda Vondra (October 2013)
Open content licensed under CC BY-NC-SA
For efficiency, we compute where is the Euler totient function , and precompute the inside sum .
In all, values of the Mertens function are plotted as red bars in intervals of 100 per visual frame .
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 .
For , , suggesting observable patterns continuing to this range.
In , 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.
 Wikipedia. "Möbius Function." (Jun 17, 2013) en.wikipedia.org/wiki/Mobius_function.
 Wikipedia. "Mertens Function." (May 3, 2013) en.wikipedia.org/wiki/Mertens_function.
 Wikipedia. "Farey Sequence." (Jun 9, 2013) en.wikipedia.org/wiki/Farey_sequence.
 J. Vondra. "Experimenting with Sums of Primitive Roots of Unity" from the Wolfram Demonstrations Project. demonstrations.wolfram.com/ExperimentingWithSumsOfPrimitiveRootsOfUnity.
 Wikipedia. "Euler's Totient Function." (Sep 18, 2013) en.wikipedia.org/wiki/Euler’s_totient _function.
 S. Wolfram. "Mertens Conjecture" from the Wolfram Demonstrations Project. demonstrations.wolfram.com/MertensConjecture.
 L. Quet, "Sum of Positive Integers, k, where k <= n/2 and GCD(k,n)=1." (Jan 20, 2002) oeis.org/A066840.