We study the convergence of the parameter family of series:

The prime numbers are not randomly distributed but, there are random models that capture important properties of the distribution of prime numbers well (e.g., [

Now, one can construct a random walk out of these random variables: start at zero. At time

This example is known as a lacunary Fourier series, that is its frequencies fulfill the growth condition given above. Its random properties are a classical field of research. In the literature, the sequence of prime numbers

On the other hand, we can look at other manifestations of randomness in lacunary series (e.g., in the example in

However, we can show that our series

Another historical example that is non-differentiable, but multifractal, is the Riemann function

While prime sums are extensively studied in the context of the famous prime conjectures (e.g., for Vinogradov’s theorem and the like), we have not found a treatment of trigonometric series over prime numbers. The reason for this is most probably that these series do not have the necessary form to help to progress in the proofs of the prime conjectures where prime exponential sums play a dominant role. As mentioned above, these series have not been studied in the context of lacunary series as prime numbers neither grow fast enough nor have known arithmetic properties, which are necessary for a straightforward analysis.

By using the results of prime number theory, we are nevertheless able to show conditions on the differentiability and self-similarity of our prime series. Experimentally, we explore also its box dimension in dependence of

There are basically two factors that influence the smoothness and convergence of a function series

The faster the coefficients

The faster the frequencies

The nature of these influences, easily deduced, are also backed by the long history of studies on the following two families of functions (and derived families):

Let:

([

On the other hand, one has the family of Riemann’s functions (whose authorship by Riemann is apparently only confirmed by Weierstrass) defined by:

([

In the following, we aim to give a similar description for our function series. Let us start with some preliminary definitions, which are necessary for what follows.

We call a function

We call the supremum of

Let

With these notation, we have the following estimation, which is a special case of Proposition 5 in [

In the spirit of the results in

We have

We take now

For the first result, we use the properties of the prime zeta function

Secondly, for any

This sequence of derivatives converges uniformly with the same argument as above for

First of all, let

From this formula and

Approximating the logarithmic integral, this implies:

If

Combining Equations (

For any

The exponent

For this step, we use a method developed by Jaffard in [

We choose a function

As the support of

Recall that we have just proven that

The gap

The graph of the function

Denote by

That is, we can decompose the partial sum into residue classes of the prime numbers and the roots of unity of cosine. One knows that the number of primes

The factors

As we have

See

Further, we compute numerically the box dimension of the graph of

In accordance with our results on the regularity of

The quite similar behavior of lacunary and random Fourier series allow us to think that it might be possible to capture the random character of the series

The terms

(Philipp-Stout [

While the Hadamard growth condition (

Because of the reason mentioned above, we have not been able to show the central limit theorem for the random variables

The properties of the series

Although the series might be reminiscent of the Riemann zeta function or other number-theoretical functions, we did not construct

We would like to thank Juri Merger and Florian Pausinger for his helpful hints and questions.

D.V. conceived of and designed the experiments. D.V. and D.B. performed the experiments. D.V. and D.B. analyzed the data. D.B. wrote the paper.

The authors declare no conflict of interest.

Graph of (

Graph of (

Graph of

Graph of

Graph of

Graph of

The graph of the real part of

(

Normal distribution of

Normal distribution of