# Correlating the Mertens Function with the Farey Sequence

Correlating the Mertens Function with the Farey Sequence

The Möbius function is defined to be if is the product of distinct primes, and zero otherwise, with [1]. The values of the Möbius function are for positive integers .

μ(n)

q

(-1)

n

q

μ(1)=1

1,-1,-1,0,-1,1,…

n≥1

The Mertens function is defined to be the cumulative sum of the Möbius function, [2], so that the values of the Mertens function are for positive integers . (In this Demonstration we start the sequence at .)

M(n)

M(n)=μ(m)

∑

m≤n

1,0,-1,-1,-2,-1,…

n≥1

n≥3

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 [3]. For , the new terms are , , , , , . Therefore, =0,,,,,,,,,,,,1.

F

n

n

n

n=1,2,…,6

{0,1}

1

2

,

1

3

2

3

,

1

4

3

4

,,,

1

5

2

5

3

5

4

5

,

1

6

5

6

F

6

1

6

1

5

1

4

1

3

2

5

1

2

3

5

2

3

3

4

4

5

5

6

Truncating the Farey sequence to include only the fractions less than and omitting 0 and 1, define ={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 that describes an asymmetry in the distribution of about [4]. Then the values of are .

1

2

f

n

D(n)=8-δ

∑

δ∈

f

n

1

4

f

n

1

4

D(n)

-,-,-,-,…

2

3

2

3

22

15

4

5

This Demonstration compares the Mertens function values (red) with (yellow), and shows the difference in green. The values shown range from to .

D(n)

M(n)-D(n)

100(r-1)

100r