Fractal Strings and the Riemann Hypothesis

Spectrum of a String

The spectrum of an ordinary string is determined by the simple modes of vibration of the string

For a string of unit length, the set of wavelengths \( \lambda_i \) are given by

\[ \lambda_i = 1, \frac12, \frac13 , \frac14 \ldots = \frac1i \]

Which are the reciprocals of the natural numbers. These are called the harmonics of the string

Fractal Strings

Take a string of any length. From this string cut out several parts each of length \(\ell_i\). In general after infinitely many cuts to the string, we will be left with a fractal. We call such a construction a fractal string

An example of such is the Cantor string. To create the Cantor string, start by removing the middle thirds of a unit string. Then remove the middle thirds of the remaining string, continuing this process on forever. What we will be left with in the end is a fractal called the Cantor Set

Sound of Fractal Strings

We can use each of the lengths \(\ell_i\) cut from the string to create a stringed instrument like a harp. The spectrum of sound produced by this harp is simply a combination of the spectrum of each string

A question we can pose is the following:

Can the sounds produced by such an instrument tell us anything about the geometry of the underlying fractal, in other words, can we hear the shape of a fractal string?

Dimension of a Fractal

There are many ways to define the concept of a dimension. One interesting way of defining dimension is by zooming. If we zoom in by a factor of \(2\) onto a line, then we only see \( \frac12\) of the line. Do the same for a 2D surface, we see \( \frac14 \)of the surface, and for a 3D object we see \(\frac18 \) of the object.

Different zooms

If we zoom in by a factor of \(2\) then we see \( \left(\frac12\right)^D \) of the object. Where \(D\) is the dimension of the object. We can turn this around to give a definition that generalizes to all fractals as well

If we call the portion of an object that remains visible after zooming \( p\), then we say that the dimension of the object \( D \) is given by:

\[ D = - \log_2 p \]

We can also use zoom factor other than 2 and generalize the formula

\[ D = - \log_k p \]

where k is the zoom factor

Fractal strings usually have dimensions between \(0\) and \(1\)

When we zoom into a Cantor string with a zoom factor \( k = 3 \), only a \( \frac12 \) of the string remains visible (i.e \( p = \frac12 \))

Therefore the Cantor string has dimension \( D = -\log_3 \frac12 = log_3 2 \approx 0.6309 \)

zooming into a cantor string

Formula for the Spectrum, Formula for the Primes

A simple problem we can now pose is the following: given any fractal string can we calculate the number of lengths greater than or equal to a certain number

For example, for a cantor string we may ask how many lengths are there below \( \frac 19 \), the answer would be \(3\). \(1\) of length \( \frac13\) and \(2\) of length \(9\)

Given an arbitrary fractal string, the geometric counting function \( N_\ell (x) \) is defined to be the number of lengths greater than or equal to \( \frac1x \). In the cantor string example \( N_\ell (9) = 3 \)

The next question we might ask is whether or not there is some good way to approximate the geometric counting function. The answer is yes, for a certain subset of fractal strings called Minkowski measurable, the geometric counting function can simply be approximated by \(N_\ell (x) = \mathfrak Mx^D \), where \(D\) is the dimension, and \(\mathfrak{M}\) is a quantity called the Minkowski content of the fractal

Yet another counting function that will be of interest to us is the spectral counting function \( N_f (x) \) counts the number of wavelengths greater than or equal to \( \frac1x\)

Or equivalently, since frequency is the reciprocal of wavelength (assuming wave speed of 1 unit), we may count the number of frequency modes below \(\)

For an ordinary string of length \(L\), the answer is simply given by \( N_f(x) = xL\)

Another counting function, which at first seems unrelated to the other two, is the prime counting function \( N_p(x) \). It counts the number of primes less than \(x\). The approximation, based on the prime number theorem, is given by \( N_p(x) = x\ln x \)

Riemann was able to find an improved approximation using his zeta function

Zeta Functions

The original Riemann zeta function \( \zeta (s) \) can be defined by summing over the harmonics of the unit string each raised to the variable of \( s\)

\[ \zeta(s) = \sum\limits_{i=1}^\infty \left( \frac1i \right)^s \]

It is possible to generalize the zeta function to sum over any spectrum of values. For example if we choose as our basis the lengths of the lines in the fractal string, then we get the geometric zeta function \( \zeta_\ell (s) \)

\[ \zeta_\ell (s) = \sum (\ell_i)^s \]

For the geometric zeta function taking as our basis the lengths \( \ell_i \) of our fractal string

The spectral zeta function is formed from the set of all wavelengths in the spectrum of the fractal string.

\[ \zeta_\lambda = \sum\limits_{i=0}^\infty (\lambda_i)^s \]

The spectral zeta function for a unit string is the same as the Riemann zeta function

Since fractal strings are formed from strings of different lengths \( \ell_i\) each producing a spectrum \(\frac \ell i \), it follows that the spectral zeta function is the product of Riemann zeta function and the geometric zeta function

\[ \zeta_\lambda(s) = \zeta(s) \zeta_\ell (s) \]

Complex Dimension

The geometric zeta function always diverges for \(\zeta_\ell(D) \), where \(D\) is the dimension of the fractal string

For a general fractal string there are other points where the geometric zeta function also diverge, these are called the complex dimensions of the fractal string

Complex dimensions allow us to generalize the notion of a dimension. Earlier we defined the dimension of a fractal by zooming into it. It turns out that as we zoom into a typical fractal we see more details. The complex dimensions of a fractal string encode precisely such details

To take a concrete example, let us once again look at the cantor string. Earlier we showed that the dimension of the cantor string is \(log_3 2 \approx 0.6209 \). But it turns out that if we allow for complex numbers, then there are infinitely many solutions to \( log_3 2 \), these are the complex dimensions of the cantor string

Just like the regular (real) dimension, the complex dimensions also contain information about what happens when you zoom in

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