On the Growth of Some Functions Related to z ( n )

: The order of appearance z : Z > 0 → Z > 0 is an arithmetic function related to the Fibonacci sequence ( F n ) n . This function is deﬁned as the smallest positive integer solution of the congruence F k ≡ 0 ( mod n ) . In this paper, we shall provide lower and upper bounds for the functions ∑ n ≤ x z ( n ) / n , ∑ p ≤ x z ( p ) and ∑ p r ≤ x z ( p r ) .


Introduction
Perhaps the most important of the binary recurrences is the Fibonacci sequence (F n ) n . This sequence starts with F 0 = 0 and F 1 = 1 and it satisfies the 2nd order recurrence relation F n+2 = F n+1 + F n (for n ≥ 0). A well-known, explicit, formula for the nth Fibonacci number is called the Binet-formula where α := (1 + √ 5)/2 and β := (1 − √ 5)/2. It follows from this formula that the estimates α n−2 ≤ F n ≤ α n−1 , hold for all n ≥ 1.
The study of the divisibility properties of Fibonacci numbers has always been a popular area of research. For example, it is still an open problem to decide if there are infinitely many primes in that sequence. In order to study such kind of Diophantine problems, the arithmetic function z : Z >0 → Z >0 was defined by setting z(n) = min{k ≥ 1 : n | F k }. This function is called the order of appearance in the Fibonacci sequence. For more results on z(n), see [1] and references therein.
In 1878, Lucas ([2], p. 300) established that z(n) is well defined and, in 1975, J. Sallé [3] proved that z(n) ≤ 2n, for all positive integers n. This is the sharpest upper bound for z(n), since for example z(n) = 2n if and only if n = 6 · 5 k , for k ≥ 0. (1) However, apart from these cases this upper bound is very weak. For instance, z(2255) = 20 < 10 −2 · 2255. In fact, Marques [4] gave sharper upper bounds for z(n) for all positive integers n = 6 · 5 k . These upper bounds depend on the number of distinct prime factors of n, denoted by ω(n).
In the main stream of the Analytic Number Theory, we have the three following functions where Λ(n) is the well-known von Mangoldt function defined as log p if n = p r , for some prime number p and r ≥ 1, and 0 otherwise (see, e.g., [5,6]). The functions ϑ(x) and ψ(x) are called the first and the second Chebyshev functions, respectively. Note that ψ(x) can be rewritten as ∑ p r ≤x log p.
Here (and in all what follows) ∑ n≤x , ∑ p≤x and ∑ p r ≤x mean that the sum is taken over all positive integers, all prime numbers and all prime powers belonging to the interval [1, x], respectively. Probably, the main importance of the functions ψ and ϑ relies in the proof of the celebrated Prime Number Theorem which states that where π(x) = ∑ p≤x 1 is the prime counting function. Indeed, the prime number theorem and the statements ϑ(x) ∼ x and ψ(x) ∼ x are all equivalent. Here f (x) ∼ g(x) (asymptotic equivalence) means that f (x)/g(x) tends to 1 as x → ∞ (in another way, ) means a function h(x) with lim x→∞ h(x)/g(x) = 0). Actually, one has the following stronger fact (2) Here we shall use the Landau symbols in their usual meaning, i.e., we say that g means that f g and g f . Another function of great interest is the harmonic function H(x) whose image for x ∈ Z >0 is called the xth harmonic number and denoted by H x . These numbers gained much attention with their relation to the Riemann hypothesis. In fact, the Riemann hypothesis is equivalent to prove that d(n) ≤ H n + e H n log H n , for all n ≥ 1, where d(n) is the sum of the positive divisors of n (see [7]). We observe that the harmonic series, i.e., lim x→∞ H(x) is a well-studied example of divergent series. In fact, it holds that which agrees with its very slow divergence.
In this paper, we are interested in studying the growth of the following Fibonacci versions of H(x), ϑ(x) and ψ(x), thus, the functions Z H (x), Z ϑ (x) and Z ψ (x) (see Figure 1), for a positive real x, which are defined as First, observe that since 1 ≤ z(n) ≤ 2n, then the following trivial estimates hold However, we found the previous bounds by neglecting the contribution of z(n) (which is much bigger than 1 and much smaller than 2n, in almost all cases). In fact, by taking z(n) into account, we obtain Again, with an extra effort, we can improve this by proving that Since the number of prime powers in [1, x] is bigger than π(x), a similar direct inequality (that one for Z ϑ (x)) could be derived for Z ψ (x). However, by using the behavior of z(p r ), we can obtain better estimates such as Theorem 3. We have that Note that even with a larger number of possibilities in the sum of Z ψ (x), its bounds are the same (in order) than the ones for Z ϑ (x) (Theorem 2). The explanation for this, follows from the fact that the contribution, i.e., the number of powers of p (for example) belonging to [1, x] In other words, this amount is almost negligible (compared with x, in terms of order).
In a few words, the proof of the results combine some new (sharper upper bounds for z(n) due to Marques) and classical results (such as results due Abel, Sathé, Selberg) in Number Theory.

Auxiliary Results
In this section, we shall present some tools which will be very useful in the proofs. We start with some results due to Marques [4], which will be very helpful in our proof. Thus, we shall state his results as lemmas (in what follows, the 2-adic valuation of n is ν 2 (n) = max{k ≥ 0 : 2 k | n}).
where, as usual, ( a q ) denotes the Legendre symbol of a with respect to a prime q > 2.
Lemma 2. Let n be an odd integer number with ω(n) ≥ 2, then Lemma 3. Let n be an even integer number with ω(n) ≥ 2, it holds that if ω(n) = 2 and 5 | n; 2n, if ω(n) = 2 and 5 n; if ω(n) = 2 and 5 | n; n, if ω(n) = 2 and 5 n; The next lemma is a powerful result in analytic number theory which is related to positive integers with fixed number of distinct prime factors.

Lemma 4 (Sathé-Selberg Formula). For any positive constant A, we have
In the previous statement Γ(z) = ∞ 0 x z−1 e −x dx (for x > 0) is the well-known Gamma function. The proof of Lemma 4 can be found in [8,9]. Our last tool is a very useful formula due to Abel which makes an interplay between a discrete sum and an integral (continuous sum). More precisely, Lemma 5 (Abel's Summation Formula). Let (a n ) n be a sequence of real numbers and define its partial sum A(x) := ∑ n≤x a n . For a real number x > 1, let f be a continuously differentiable function on [1, x]. Then ∑ n≤x a n f (n)

Remark 1.
We remark that, throughout what follows, the implied constants in and can be made explicit. Here, we decided to use asymptotic bounds in order to leave the text more readable. However, we shall provide the explicit inequalities for convenience of the reader (they can be found in [10], for example).
As usual, from now on we use the well-known notation [a, b] = {a, a + 1, . . . , b − 1, b}, for integers a < b. Now we are ready to deal with the proof of our results.

The Proof of Theorem 1
Since, by definition, n | F z(n) , then n ≤ F z(n) ≤ α z(n)−1 and so z(n) > log n/ log α. Thus Now, we shall use Lemma 5 for a n = 1/n and f (x) = log x. Then Since H(x) = log x + O(1) and and so Z H (x) (log x) 2 . For the second part, we use Lemmas 1, 2 and 3 to derive that for all n > 1. First, let us write Z H (x) as where h(x) = max{ω(t) : t ≤ x}. By using that z(n) ≤ 7 · (2/3) ω(n) n, we have which can be written as Now, we shall use Lemma 4 to deal with the first sum in the right hand side above. Since G(z) converges uniformly and absolutely in any bounded set, we have max z∈[0,1] {|G(z)|} ≤ C, for some positive constant C. Now, by Lemma 4 for A = 1, we get |G(z k )| ≤ C (for z k := (k − 1)/ log log x < 1) and For the second sum in the right hand side of (6), we use that #P k (x) ≤ x to obtain where we used that log log x + 1 > log log x. Since 3/2 > 3 √ e, then By combining (6), (7) and (8), we obtain the desired result.

The Proof of Theorem 2
By the Prime Number Theorem, we have that ϑ(x) ∼ x. In particular, it holds that ϑ(x) x. Since ϑ(x) = ∑ p≤x log p, then where we used that z(p) > log p/ log α. For the second part, since z(p) ≤ p + 1 ≤ 3p/2, then

The Proof of Theorem 3
Note that, by Theorem 2, we have x.
For the second part, since there exist exactly log x/ log p powers of p in the interval [1, x], we can write Z ψ (x) as By using Lemma 1 (ii), we get x. Then, which completes the proof.

Conclusions
In this paper, we study some problems related to the order (of appearance) in the Fibonacci sequence, denoted by z(n). This arithmetic function plays an important role in the comprehension of some Diophantine problems involving Fibonacci numbers (the most important one is the open problem about the existence of infinitely many Fibonacci prime numbers). The problems are related to the growth of Fibonacci versions of well-known number-theoretic functions (related to the Prime Number Theorem) like the first and second Chebyshev functions, ϑ(x) = ∑ p≤x log p and ψ(x) = ∑ p r ≤x log p and the harmonic function H(x) = ∑ n≤x 1/n. These Fibonacci-like functions are defined as Z ϑ (x) = ∑ p≤x z(p), Z ψ (x) = ∑ p r ≤x z(p r ) and Z H (x) = ∑ n≤x z(n)/n. In particular, we shall find effective bounds for these three functions. The proofs combine elementary facts related to z(n) (such as Marques' upper bounds) together with some deep tools from Analytic Number Theory (such as Abel's summation formula and Sathé-Selberg formula).