Many mathematicians have been interested in the study of recursive sequences. Among them, a class of “chaotic” sequences are named “meta-Fibonacci sequences.” The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the

The set of the integer sequences, denoted by

There are classes of sequences which are not linear (like the Fibonacci sequence). For instance, the sequence

Probably the first class of recursive sequences without a fixed order, was proposed, in 1979, by Hofstadter and Gödel [

At first glance, the “self-definition” of

Many mathematicians worked on conditional results related to the

By knowing the growth of

In 2017, Alkan et al. [

Furthermore, we recall the

A few values of this function are

This function plays an important role in number theory and combinatorics (mainly due its unpredictable behavior). Its

The

We point out the stronger relation between

The

The

Recall that, we say

In the same spirit than by Alkan et al. [

Throughout the paper, we shall use the familiar notation

Here, we intend to consider the behavior of the accumulation function of

Let

A few values of the sequence (_{Q}(_{n≥1} are

We note the apparently chaotic behavior of

Also, its structure seems to be chaotic and its shape seems to be as a growing “electrocardiogram” (see Figure

Now, we shall split our study into four points: growth, generational structure, pseudo-periodicity, and statistical viewpoint.

In all what follows, we shall suppose that

It is almost unnecessary to stress that one of the first properties to study in the direction of a better comprehension of the behavior of a “chaotic” function is its growth (for large time). In fact, the structure of such a function is strongly reflected in its growth properties. For this reason, this section will be devoted to this kind of study.

By the Hypothesis (H) (in particular,

However, this upper bound can be sharpened with the aid of some analytic number theory facts. For example, we know that

Let

Before applying the previous formula, observe that

Note that, by the properties of Landau’s symbols,

By the prime number theorem, we have

Again, by the properties of Landau’s symbols, we have

It holds that

We remark that since the Riemann Hypothesis (RH) is equivalent to

By assuming that the Riemann Hypothesis is true, then, for all

Note that in Figure

Graph of

The growth behavior of the graphical structure of

Graph of

In an extensive empirical/heuristical study (see Figure

For all positive integers

In the next section, we shall see that a kind of periodicity of

We point out that most of these findings are empirical observations, since virtually speaking, nothing has been proved rigorously about the

A function

Clearly, the Hofstadter–Mertens function is aperiodic. However, we can define another kind of periodicity.

A function

The Hofstadter–Mertens function is meta-periodic.

For instance, for some values of

For

For

For

Now, we wish to study the behavior of

Now, we define

The scatter plot of

Figure

Standard deviation, mean, and median of _{Q}(n).

We close this study by comparing

The zeta function is only one example of the called

Many properties of prime numbers are encoded by Dirichlet’s series and its Euler’s product. For example, for the Riemann zeta function, we have

We have that

Here, as usual,

In order to prove this fact, we recall the

Let

A well-known fact is that

In this paper, we continue the fruitful program started by Hofstadter, Golomb, Pinn, Alkan, Fox, and Aybar (among others) to study the behavior of meta-Fibonacci sequences, mainly the _{Q}(

The data calculated by the software Mathematica for Figures 1, 2, 3, 4, 6, 7, and 8 used to support the findings of this study are included within the supplementary information files.

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author would like to thank Faculty of Science, University of Hradec Králové, the project of Excellence PrF UHK No. 2213/2020, for the support.

The data of Figures 1, 2, 3, 4, 6, 7, and 8 are in separate files Figure 1.dat, Figure 2.dat, Figure 3.dat, Figure 4.dat, Figure 6.dat, Figure 7.dat, and Figure 8.dat.