Nonlinear Dynamics of Complex SystemsView this Special Issue
On Some Properties of the Hofstadter–Mertens Function
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 Q-sequence. Recently, Alkan–Fox–Aybar and the author studied the pattern induced by the connection between the Q-sequence and other known sequences. Here, we continue this program by studying a “Mertens’ version” of the Hofstadter sequence, defined (for ) by , where µ(n) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to “meta-sequences”.
The set of the integer sequences, denoted by , is in the main stream of the mathematical studies. For example, the problems associated with the set of prime numbers are central topics in mathematics and in many recent applications (including modern cryptography) are based on these sequences. Another very famous numerical sequence is the sequence of Fibonacci numbers defined by the recurrence (for ), with initial values . The Fibonacci sequence is a binary recurrence; i.e., each term is the sum of the two preceding ones. Binary means order 2, so for an order k sequence, we need to know the previous k terms (in some combinations).
There are classes of sequences which are not linear (like the Fibonacci sequence). For instance, the sequence defined by the quadratic recurrence , with (where c is a given complex number), is a standard example of fractal sequence (connected to Mandelbrot’s set). However, besides being nonlinear, it still has order 1.
Probably the first class of recursive sequences without a fixed order, was proposed, in 1979, by Hofstadter and Gödel . In fact, they defined by the self-recurrence relation as follows:with initial values . The term “self-recurrence” is the key point here. Because we must first know the value of in order to calculate the value of , the prefix “meta” means that this kind of sequence transcends, in some sense (maybe because of the nonexistence of a fixed order), the usual examples (i.e., it is “beyond” the standard recurrent sequences). The first terms of areand the graph of in the interval is plotted in Figure 1.
At first glance, the “self-definition” of Q(n) appears to be a very strange definition. Paradoxically, so far, we do not know even if Q(n) exists for all positive integers n (this is confirmed for ) ).
Many mathematicians worked on conditional results related to the Hofstadter sequence (i.e., ) under the assumption of its well definition. For example, Golomb  was the first to prove that , provided that this limit exists. A much weaker version of this result would be enough to ensure that the Hofstadter sequence is well defined. In fact, it suffices that , for all (the limit says that grows as ).
By knowing the growth of , namely, , Pinn  developed the study of generations by paying attention to the -graph. For him, these generations are the intervals such that the partial graph contains a complete “sausages” pattern as in Figure 1.
In 2017, Alkan et al.  made a very interesting discovery by studying the sequence (named by them as Hofstadter chaotic heart sequence, see its “heart fractal-like structure” in Figure 2), where C(n) is the Hofstadter–Conway $10,000 sequence is defined by the following recurrence:with initial values (in contrast with , and it was proved that this sequence is well defined on ).
Furthermore, we recall the Möbius function which is defined by
A few values of this function are
This function plays an important role in number theory and combinatorics (mainly due its unpredictable behavior). Its accumulation (or summation) function denoted by (and called the Mertens function) is defined by (here the sum is taken over all positive integers smaller or equal to ).
The Mertens function slowly grows in positive and negative directions both on the average and in peak value, oscillating in an apparently chaotic manner passing through zero when has the following values:
We point out the stronger relation between and some important number theoretic statements:(i)The Prime Number theorem (which says that tends to 1 as ) is equivalent to (here is the prime counting function, namely, the number of prime numbers belonging to )(ii)The Riemann Hypothesis is equivalent to , for all (see [6, Theorem 14.25]), where, as usual, and are the standard Big-O and Little-o Landau notations
Recall that, we say if there exists a positive constant , such that for all sufficiently large (the same meaning as and if ). Also, we denote , if tends to 1 as ( and are said to be asymptotically equivalent).
In the same spirit than by Alkan et al. , in a very recent paper, the author of  studied the relationship between the functions and , by defining . In this paper, we continue this program by introducing and analyzing the accumulation function of which gives relations between meta-Fibonacci sequences, prime factorization, and random walks.
Throughout the paper, we shall use the familiar notation , for integers .
2. Hofstadter–Mertens Function
Here, we intend to consider the behavior of the accumulation function of , denoted by , which we call Hofstadter–Mertens function and it is defined as follows.
Definition 1. Let , where denotes the nth term of Hofstadter sequence and μ(n) is the Möbius function.
A few values of the sequence (MQ(n))n≥1 areWe note the apparently chaotic behavior of which passes through zero (i.e., and ) when x takes the following integer values:Also, its structure seems to be chaotic and its shape seems to be as a growing “electrocardiogram” (see Figure 3).
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 is well defined and that tends to as . We shall quote this as “Hypothesis (H).”
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, ) together with the fact that , we deduce that
However, this upper bound can be sharpened with the aid of some analytic number theory facts. For example, we know that and by Hypothesis (H), we can write . This allows us to invoke a very useful formula due to Abel which makes an interplay between a discrete sum and an integral (continuous sum).
2.2. Abel’s Summation Formula
Let be a sequence of real numbers, and define its partial sum . For a real number , let be a continuously differentiable function on . Then,
Before applying the previous formula, observe that can be rewritten as
Note that, by the properties of Landau’s symbols, and now, in order to obtain an estimate to , we are in the position to apply Abel’s Summation Formula with the choice of and . Thus,
By the prime number theorem, we have and then
Again, by the properties of Landau’s symbols, we have and so we arrive at the following fact.
Fact 1. It holds thatWe remark that since the Riemann Hypothesis (RH) is equivalent to , then, by proceeding along the same lines as before, we deduce the following fact.
Fact 2. By assuming that the Riemann Hypothesis is true, then, for all , it holds that
Remark 1. Note that in Figure 4, the bound (red colored) seems to be very huge as compared to . However, we point out that a similar feeling happens by plotting a similar graphic to and . In fact, this leads to the “very probable” conjecture raised by Mertens: , for all . However, this conjecture was proved to be false (see ) for some (nonexplicit) counter examples of astronomical order about .
2.3. Generational Structure
The growth behavior of the graphical structure of brings a complex fractal-like structure. In fact, these kinds of patterns are commonly called “generational structure” of a meta-Fibonacci sequence (for more information about these structures for other meta-Fibonacci sequences, we refer the reader to [9–11]). In our case, these “generations” are the repeated “zigzag” pattern. More precisely, with Pinn’s terminology, it is possible to partition the set of positive integers aswhere is a finite interval of natural numbers which is known as the generation of the sequence. In our case, each generation will be in such a way that (see Figure 5)(i)(ii)
In an extensive empirical/heuristical study (see Figure 6), we were not able to find a pattern for and . The main reason may lie in the chaotic behavior of (related to random walks, for example).However, it is possible to deduce that can be made arbitrarily large. That is, we have the following fact.
Fact 3. For all positive integers , there is an interval with length strictly larger than .
In the next section, we shall see that a kind of periodicity of is responsible for the veracity of this fact.
We point out that most of these findings are empirical observations, since virtually speaking, nothing has been proved rigorously about the Q-sequence, so far (as previously mentioned).
A function is said to be periodic if there exists a positive integer such that , for all (in other words, the function repeats its values in regular intervals or periods). The most important examples of periodic functions are the trigonometric functions, which repeat over intervals of radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.
Clearly, the Hofstadter–Mertens function is aperiodic. However, we can define another kind of periodicity.
Definition 2. A function is said to be meta-periodic if for any positive integer , there exist infinitely many positive values of , such that . Let T be a positive integer, and let be the first T prime numbers. We have that the congruence systemhas infinitely many solutions in (in a residue class modulo ), say , by the Chinese remainder theorem. Clearly, for a particular solution , we have that and so , for all . Thus, one has . In particular, we havefor all . Thus, we have the following fact.
Fact 4. The Hofstadter–Mertens function is meta-periodic.
For instance, for some values of T, we have the following:(i)For , we have the family of solutions (ii)For , we have the family of solutions (iii)For , we have the family of solutions
2.5. Statistical Viewpoint
Now, we wish to study the behavior of in a statistical vein. For this, our method will be based on a process that appeared in OEIS A283360 (see also its Link section) for the behavior that keeps the main characteristic of Q-sequence with deviations of noise in generations. Here, we define as the remainder after division of by . So, for , attains the following values:
Now, we define and Figure 7 is the scatter plot of and for from 1 to 10000.
Figure 8 shows the evaluation (in a statistical viewpoint) of the standard deviation (SD), mean (arithmetic mean), and median of values of in the interval , for .
3. Hofstadter–Mertens Function × Riemann Zeta Function
We close this study by comparing and the Riemann zeta function (for ). We know that ζ(s) converges for all and admits an analytic continuation (via Abel Summation Formula) for except for a simple pole at , with residue 1 (in fact, Riemann extended this continuation for all complex planes but ).
The zeta function is only one example of the called Dirichlet series which, for an arithmetic function , is defined by
Many properties of prime numbers are encoded by Dirichlet’s series and its Euler’s product. For example, for the Riemann zeta function, we havewhere the product is taken over all prime numbers. We know the huge importance of primes in mathematics and even in real life (as in cryptography). So, among the attempts made in this direction, we were able to provide the following fact.
Fact 5. We have thatwhereHere, as usual, denotes the number of distinct prime factors of and is the Euler totient function.
In order to prove this fact, we recall the Dirichlet convolution between two Dirichlet’s series.
Definition 3. Let f and be arithmetic functions; then, the Dirichlet convolution of and , denoted by is defined byA well-known fact is that . Thus, let us consider (for all ), i.e., and then,where . Since, by Hypothesis (H), , then . Note thatwhere (where is a prime) is called the radical of n (the proof of (26) follows from the fact that the left-hand side and the right-hand side, in formula (26), represent arithmetic multiplicative functions and so it is enough to compare them when is a prime power). Now, we use to conclude Fact 5.
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-sequence. Here, we studied the sequence which is defined as the accumulation function of the product between the Q-sequence and the Möbius function (we call as Hofstadter–Mertens function). The sequence MQ(n) is studied with emphasis on its chaotic behavior. We split the text into four parts. We started by growing properties of and its relation with the Riemann hypothesis (here, we used some analytic tools). Then, we present some data regarding its “generational structures” together with some other facts. In the third part, we worked on the pattern repetition of by showing that it satisfies a kind of “meta-periodicity.” We finish by mentioning some statistical viewpoints of the Hofstadter–Mertens function, such as its mean, median, and standard deviation (in a large scale). In the final section, we still present some theories of Dirichlet series related to which could have some theoretical interest in detection of primes or problems related to the Riemann zeta function.
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.
Conflicts of Interest
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. (Supplementary Materials)
D. R. Hofstadter and E. Gödel, Bach: An Eternal Golden Braid, Basic Books, New York, NY, USA, 1979.
S. W. Golomb, Discrete Chaos: Sequences Satisfying “Strange” Recursions, 1991, https://oeis.org/A005185/a005185 1.pdf, In press.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, University of Oxford, Oxford, UK, 1951.