Correlating the Mertens Function with the Farey Sequence

r

1

The Möbius function

μ(n)

is defined to be

q

(-1)

if

n

is the product of

q

distinct primes, and zero otherwise, with

μ(1)=1

[1]. The values of the Möbius function are

1,-1,-1,0,-1,1,…

for positive integers

n≥1

.

The Mertens function

M(n)

is defined to be the cumulative sum of the Möbius function,

M(n)=

∑

m≤n

μ(m)

[2], so that the values of the Mertens function are

1,0,-1,-1,-2,-1,…

for positive integers

n≥1

. (In this Demonstration we start the sequence at

n≥3

.)

The Farey sequence

F

n

of order

n

is the set of irreducible fractions between 0 and 1 with denominators less than or equal to

n

, arranged in increasing order [3]. For

n=1,2,…,6

, the new terms are

{0,1}

,



1

2



,



1

3

,

2

3



,



1

4

,

3

4



,



1

5

,

2

5

,

3

5

,

4

5



,



1

6

,

5

6



. Therefore,

F

6

=0,

1

6

,

1

5

,

1

4

,

1

3

,

2

5

,

1

2

,

3

5

,

2

3

,

3

4

,

4

5

,

5

6

,1

.

Truncating the Farey sequence to include only the fractions less than

1

2

and omitting 0 and 1, define

f

n

={p/q:p/q<1/2,gcd(p,q)=1,1≤p<n/2,3≤q≤n}

. Define a measure of the Farey sequence distribution by

D(n)=8

∑

δ∈

f

n

1

4

-δ

that describes an asymmetry in the distribution of

f

n

about

1

4

[4]. Then the values of

D(n)

are

-

2

3

,-

2

3

,-

22

15

,-

4

5

,…

.

This Demonstration compares the Mertens function values (red) with

D(n)

(yellow), and shows the difference

M(n)-D(n)

in green. The values shown range from

100(r-1)

to

100r

.