A function of the form
       az + b
f(z) = ------ ,
       cz + d
where the parameters a,b,c,d are complex numbers or real numbers or even integers (depending on the application).

There are lots of useful properties of möbius functions. Among the places where möbius functions find application are complex analysis (constraining the parameters appropriately yields the holomorphic automorphisms of the unit disk), continued fractions (where they exactly describe the operations used in constructing a continued fraction), computer arithmetic (due to their connection to continued fractions) and hyperbolic geometry of the plane (for the same reason as their use in complex analysis: they are the isometries of the hyperbolic plane when constrained appropriately).

There appears to be yet another, very different kind of Möbius function μ(x) that is important in analytical number theory and the Riemann hypothesis. It is a fairly cumbersome function to define, namely:

  • It is a function over the natural numbers
  • μ(1) = 1
  • μ(x) = 0 if x has a square factor.
  • μ(x) = -1 if x is prime or the product of an odd number of distinct prime numbers.
  • μ(x) = 1 if x is the product of an even number of distinct primes.

However, there is a fairly elegant identity that relates it to the Riemann zeta function, and is derived as follows. First, we begin with the infinite product representation (the Euler product formula) of the Riemann zeta function:

ζ(s) = (1 - 2-s)-1(1 - 3-s)-1(1 - 5-s)-1(1 - 7-s)-1(1 - 11-s)-1...

The zeta function is here represented as an infinite product over all the prime numbers. We then take its reciprocal:

1/ζ(s) = (1 - 2-s)(1 - 3-s)(1 - 5-s)(1 - 7-s)(1 - 11-s)...

We then try to expand the infinite product by multiplying it out, and after some complicated contortions we see that the infinite sum that results has the following form:

1 - 1/2s - 1/3s - 1/5s - 1/7s + ...

One minus the sum of the reciprocals of all the primes raised to s, plus:

1/6s + 1/10s + 1/10s + 1/14s + ...

the sum of the reciprocals of the product of every pair of distinct prime numbers. The series goes on in this way forever, and thus consists of terms that are all the negative reciprocal raised to the s power of every natural number that is the product of an odd number of distinct primes, and the reciprocal raised to the s power of every natural number that is the product of an even number of primes:

1 - 1/2s - 1/3s - 1/5s + 1/6s - 1/7s + 1/10s + ...

This leads us straight to this elegant little identity:

  1       μ(n)
 ____ = Σ ___
 ζ(s)   n   s

The Möbius function and this identity in particular actually play a key role in the development of the theory behind the Riemann hypothesis, and its role should be further explained there.

The cumulative value of the Möbius function, M(k) = μ(1) + μ(2) + ... + μ(k) is known as Merten's function and seems to be a fairly irregular function that has characteristics similar to a random walk. The only obvious characteristic of Merten's function is that it is an increasing function, but other than that, there is nothing obvious about its features. Because of its intimate relationship with the Riemann zeta function however, the question of the rate at which it increases asymptotically is intimately connected with the truth or falsity of the Riemann hypothesis. In fact, if one could prove or disprove this bound:

M(k) = O(k1/2 + ε)

for every ε no matter how small, then one has proved or disproved the Riemann hypothesis.

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