I – Convergence of Arithmetical Functions

Let n > 1 be an integer with its canonical representation, n ¼ p α 1 1 p α 2 2 ⋯ p α k k . Put H n ð Þ ¼ max α 1 , … , α k f g , h n ð Þ ¼ min α 1 , … , α k f g , ω n ð Þ ¼ k , Ω n ð Þ ¼ α 1 þ ⋯ þ α k , f n ð Þ ¼ Q d ∣ n d and f ∗ n ð Þ ¼ f n ð Þ n . Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I d – convergence, where I d is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I – convergence of the well-known arithmetical functions, where I ¼ I q ð Þ c ¼ A ⊂  : P a ∈ A a (cid:2) q < þ ∞ (cid:2) (cid:3) is an admissible ideal on  such that for q ∈ 0, 1 ð i we have I q ð Þ c ⊊ I d , thus I q ð Þ c – convergence is stronger than the statistical convergence ( I d – convergence).

n . Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I d -convergence, where I d is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I-convergence of the well-known arithmetical functions, where is an admissible ideal on  such that for q ∈ 0, 1 ð i we have I q

Introduction
The notion of statistical convergence was introduced independently by Fast and Schoenberg in [1,2], and the notion of I-convergence introduced by Kostyrko et al. in the paper [3] coresponds to the natural generalization of statistical convergence (see also [4] where I-convergence is defined by means of filterthe dual notion to ideal). These notions have been developed in several directions in [5][6][7][8][9][10][11][12][13][14][15][16][17][18] and have been used in various parts of mathematics, in particular in Number Theory and Ergodic Theory, for example [15,[19][20][21][22][23][24][25][26][27][28] also in Economic Theory [29,30] and Political Science [31]. Many authors deal with average and normal order of the wellknown arithmetical functions (see [20,21,23,24,26,28,32,33] and the monograph [34] for basic properties of the well-known arithmetical functions). In what follows, we shall strengthen these results from the standpoint of I -convergence of sequences, mainly by I q ð Þ c -convergence and I u -convergence. On connection with that we can obtain a good information about behaviour and properties of the wellknown arithmetical functions by investigating I -convergence of these functions or some sequences connected with these functions. Specifically in [28] by means of I d -convergence, there is recalled the result that normal order of Ω n ð Þ or ω n ð Þ respectively is log log n. We managed to completely determine for which q ∈ 0, 1 ð i the sequences Ω n ð Þ log log n and ω n ð Þ log log n are I q ð Þ c -convergent. As consequence of our results, we have that the above sequences are I d -convergent to 1, what is equivalent that normal order of Ω n ð Þ or ω n ð Þ respectively is log log n. Further in [26], there is proved that the sequence log p Á a p n ð Þ log n is I d -convergent to 0 (see also [21]). We shall extend this result by means of I u -convergence of the sequence log p Á a p n ð Þ log n .
So we can get a better view of the structure of the set B ε We also want to investigate the I q ð Þ c -convergence of further arithmetical functions.

Basic notions
Let  be the set of positive integers. Let A ⊆ . If m, n ∈ , m ≤ n, we denote by A m, n ð Þthe cardinality of the set A ∩ m, n ½ . A 1, n ð Þis abbreviated by A n ð Þ. We recall the concept of asymthotic, logarithmic and uniform density of the set A ⊆  (see [35][36][37][38]).
ð Þ are called the lower and upper asymptotic density of the set A, for n ! ∞ and γ is the Euler constant, s n can be replaced by log n in the definition of δ n A ð Þ. [17,37,39,40]) and they are called lower and upper uniform density of the set A, respectively. If u A ð Þ ¼ u A ð Þ , then we denote it by u A ð Þ and it is called the uniform density of A. It is clear that for each A ⊆  we have Further densities can be found in papers [11,12]. Let n ¼ p α 1 1 p α 2 2 ⋯p α k k be the canonical representation of the integer n ∈ . Recall some arithmetical functions, which belong to our interest.
6. let p be a prime number, a p n ð Þ is defined as follows: a p 1 ð Þ ¼ 0 and if n > 0, then a p n ð Þ is a unique integer j ≥ 0 satisfying p j |n, but p jþ1 ∤n i.e., p a p n ð Þ ∥n, 7. γ n ð Þ and τ n ð Þwere introduced in connection with representation of natural numbers of the form n ¼ a b , where a, b are positive integers. Let be all such representations of given natural number n, where a i , b i ∈ . Denote by It is clear that γ n ð Þ ≥ 1, because for any n > 1 there exist representation in the form n 1 .

Ideals
A lot of mathematical disciplines use the term small (large) set from different point of view. For instance a final set, a set having the measure zero and nowhere dense set is a small set from point of view of cardinality, measure (probability) and topology, respectively. The notion of ideal I ⊆ 2 X is the unifying principle how to express that a subset of X 6 ¼ Ø is small. We say a set A ⊆ X is a small set if A ∈ I . Recall the notion of an ideal I of subsets of .
Let I ⊆ 2  . I is said to be an ideal in , if I is additive (if A, B ∈ I then A ∪ B ∈ I) and hereditary (if A ∈ I and B ⊂ A then B ∈ I ). An ideal I is said to be non-trivial ideal if I 6 ¼ Ø and  ∉ I . A non-trivial ideal I is said to be admissible ideal if it contains all finite subsets of . The dual notion to the ideal is the notion filter. A non-empty family of sets F ⊂ 2  is a filter if and only if Ø ∉ F , for each A, B ∈ F we have A ∩ B ∈ F and for each A ∈ F and each B ⊃ A we have B ∈ F (for definitions see e.g. [4,41,42]). Let I be a proper ideal in  (i.e.  ∉ I ). Then a family of sets F I ð Þ ¼ B ⊆  : there exists A ∈ I such that B ¼ nA f g is a filter in , so called the associated filter with the ideal I .
The following example shows the most commonly used admissible ideals in different areas of mathematics. Example 1.2.
a. The class of all finite subsets of  forms an admissible ideal usually denoted by I f .
b. Let ϱ be a density function on , the set g is an admissible ideal. We will use namely the ideals I d , I δ and I u related to asymptotic, logarithmic and uniform density, respectively. c. A wide class of ideals I can be obtained by means of regular non negative matrixes T ¼ t n,k ð Þ n,k ∈  (see [43]). For A ⊂  , we put d  [3,44]). Put Then I d T is a non-trivial ideal and I d T contains both I d and I δ ideals as a special case. Indeed I d can be obtained by choosing t n,k ¼ 1 n for k ≤ n, t n,k ¼ 0 for k > n and I δ by choosing n for k ≤ n, k|n and t n,k ¼ 0 otherwise we obtain I φ ideal of Schoenberg (see [2]), where φ is Euler function.
Another special case of I d T is the following. Take an arbitrary divergent series P ∞ n¼1 c n , where c n > 0 for n ∈  and put t n, d. Let μ be a finitely additive normed measure on a field S ⊆ 2  . Suppose that S contains all singletons n f g, n ∈ . Then the family In the case if μ is the Buck measure density (see [13,45]), I μ is an admissible ideal and I μ ⊊ I d .
It is easy to see, that for any q 1 , q 2 ∈ 0, 1 ð Þ, q 1 < q 2 we have The fact I c ⊊ I d in Eq. (2) follows from the following result. Let A ⊆  and P The fact that for any q 1 ,

Iand I * -convergence
The notion of statistical convergence was introduced in [1, 2] and the notion of I -convergence introduced in [3] corresponds to the natural generalization of the notion of statistical convergence.
Let us recall notions of statistical convergence, Iand I * -convergence of sequence of real numbers (see [3]). Definition 1.3. We say that a sequence x n ð Þ ∞ n¼1 is statistically convergent to a number L ∈  and we write lim stat i. We say that a sequence x n ð Þ ∞ n¼1 is I -convergent to a number L ∈  and we write I À lim In the definition of usual convergence the set A ε ð Þ is finite, it means that it is small from point of view of cardinality, A ε ð Þ ∈ I f . Similarly in the definition of statistical convergence the set A ε ð Þ has asymptotic density zero, it is small from point of view of density, A ε ð Þ ∈ I d . The natural generalization of these notions is the following, let I be an admissible ideal (e.g. anyone from Example 1.2) then for each ε > 0 we ask whether the set A ε ð Þ belongs in the ideal I . In this way we obtain the notion of the I -convergence. For the following use, we note that the concept of I -convergence can be extended for such sequences that are not defined for all n ∈ , but only for "almost" all n ∈ . This means that instead of a sequence x n ð Þ ∞ n¼1 we have x s ð Þ s ∈ S , where s runs over all positive integers belonging to S ⊆  and S ∈ F I ð Þ. Remember that I-convergence in  has many properties similar to properties of the usual convergence. All notions which are used next we considered in real numbers . The following theorem can be easily proved. Theorem 1.5 (Theorem 2.1 from [9]).
i. If I À lim x n ¼ L and I À lim y n ¼ K, then I À lim x n AE y n À Á ¼ L AE K.
ii. If I À lim x n ¼ L and I À lim y n ¼ K, then I À lim x n Á y n À Á The following properties are the most familiar axioms of convergence (see [47]).
(S) Every constant sequence x, x, … , x, … ð Þ converges to x. (H) The limit of any convergent sequence is uniquely determined. (F) If a sequence x n ð Þ ∞ n¼1 has the limit L, then each of its subsequences has the same limit.
(U) If each subsequence of the sequence x n ð Þ ∞ n¼1 has a subsequence which converges to L, then x n ð Þ ∞ n¼1 converges to L. A natural question arises which above axioms are satisfied for the concept of I -convergence. Theorem 1.6 (see [14] and Proposition 3.1 from [3], where the concept of I-convergence has been investigated in a metric space) Let I ⊂ 2  be an admissible ideal.
ii. If I contains an infinite set, then I -convergence does not satisfy (F). Theorem 1.7 (see [3]) Let I be an admissible ideal in . If I * À lim x n ¼ L then I À lim x n ¼ L.
The following example shows that the converse of Theorem 1.7 is not true. Example 1.8. Let I ¼ I  be an ideal from Example 1.2 f). Define x n ð Þ ∞ n¼1 as follows: For n ∈ D j we put x n ¼ 1 j for j ¼ 1, 2, … . Then obviously I À lim x n ¼ 0. But we show that I * À lim x n ¼ 0 does not hold.
If H ∈ I then directly from the definition of I there exists p ∈  such that In [3] was formulated a necessary and sufficient condition for an admissible ideal I under which Iand I * -convergence to be equivalent. Recall this condition (AP) that is similar to the condition (APO) in [7,35]. Definition 1.9 (see also [40]) An admissible ideal I ⊂ 2  is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets A 1 , A 2 , … f g belonging to I there exists a countable family of sets B 1 , Observe that each B j from the previous Definition belong to I . Theorem 1.10 (see [14]) From I À lim x n ¼ L the statement I * À lim x n ¼ L follows if and only if I satisfies the condition (AP).
In [44] it is proved that I d Tand I * d T -convergence are equivalent in  provided that T ¼ t n,k ð Þ n,k ∈  from Example 1.2 c) is a non-negative triangular matrix with P n k¼1 t n,k ¼ 1 for n ∈ . From this we get that I d , I δ , I φ -convergence coinside with I * d , I * δ , I * φ -convergence, respectively. On the other hand for further ideals from Example 1.2 e.g. I u , I  and I μ , respectively, we have that they do not fulfill the assertion that their I -convergence coincides with I * -convergence. Since these ideals do not fulfill condition (AP) (see [13,38,40]).
The following Theorem shows that also for all ideals I q ð Þ c for q ∈ 0, 1 ð i the concepts Iand I * -convergence coincide. Proof. It suffices to prove that for any I q ð Þ c , q ∈ 0, 1 ð i and any sequence x n ð Þ ∞ n¼1 of real numbers such that I q Therefore there exists an infinite sequence n 1 < n 2 < ⋯ < n k < ⋯ of integers such that for every k ¼ 1, 2, … X a > nk a ∈ Ak a Àq < 1 2 k : Let m k > n k 0 . Then m k belongs to some interval n j , n jþ1 À where j ≥ k 0 and does not belong to A j j ≥ k 0 ð Þ. Hence m k belongs to nA j , and then |x m k À L| < ε for every m k > n k 0 , thus lim k!∞ x m k ¼ L. Corollary 1.12 Ideals I q ð Þ c for q ∈ 0, 1 ð i have the property (AP). It is easy to prove the following lemma. Lemma 1.13 (see [3]). If I 1 ⊆ I 2 then the statement I 1 À lim x n ¼ L implies I 2 À lim x n ¼ L.

I -convergence of arithmetical functions
We can obtain a good information about behaviour and properties of the wellknown arithmetical functions by investigating I -convergence of these functions or some sequences connected with these functions. Recall the concept of normal order. Definition 1.14. The sequence x n ð Þ ∞ n¼1 has the normal order y n À Á ∞ n¼1 if for every ε > 0 and almost all (almost all in the sense of asymptotic density) values n we have 1 À ε ð Þy n < x n < 1 þ ε ð Þy n . Schinzel and Šalát in [28] pointed out that one of equivalent definitions to have the normal order is as follows. The sequence x n ð Þ ∞ n¼1 has the normal order y n À Á ∞ n¼1 if and only if I d À lim x n y n ¼ 1. The results concerning the normal order will be formulated using the concept of statistical convergence, which coincides with I dconvergence. For equivalent definitions of the normal order and more examples concerning this notion see [34,38,48]. In the papers [21, 27,28] and in the monograph [38] there are studied various kinds of convergence of arithmetical functions which were mentioned at the beginning. The following equalities were proved in the paper [28] by using the concept of the normal order.
Similarly for the functions f n ð Þ and f * n ð Þ. In [27] it is proved the following equality: Let us recall one more result from [26], let p be a prime number, there was proved that the sequence log p Á c -convergent to 0 for q ¼ 1 and it is not I q ð Þ cconvergent for all q ∈ 0, 1 ð Þ, as it was shown in [21]. In [19] it was proved that this sequence is also I u -convergent.
The following theorem shows that the assertions using the notion I u instead of I q ð Þ c , q ∈ 0, 1 ð i need to use a different technique for their proofs. First of all we recall a new kind of convergence so called the uniformly strong ℓ-Cesàro convergence. This convergence is an analog of the notion of strong almost convergence (see [6]). Definition 1.15. A sequence x n ð Þ ∞ n¼1 is said to be uniformly strong ℓ-Cesàro conver- The following Theorem shows a connection between uniformly strong ℓ-Cesàro convergence and I u -convergence. Theorem 1.16 (see [6]). If x n ð Þ ∞ n¼1 is a bounded sequence, then x n ð Þ ∞ n¼1 is I uconvergent to L if and only if x n ð Þ ∞ n¼1 is uniformly strong ℓ-Cesàro convergent to L for some ℓ, 0 < ℓ < ∞.
The sequence log p Á a p n ð Þ log n ∞ n¼2 is I u -convergent to zero i.e. for arbitrary ε > 0 the set A ε ð Þ ¼ n ∈  : log p Á a p n ð Þ log n ≥ ε > 0 n o has uniform density equal to zero. Theorem 1.17 (see [19] . This immediately implies that p α 0 ≤ N < p α 0 þ1 . Then for all n ∈ k, k þ N ð we have a p n ð Þ ¼ α < α 0 with the possible exception of one n 1 ∈ k, k þ N ð for which we could have a p n 1 ð Þ ¼ α 1 > α 0 . Assume that there exist two such numbers n 1 , n 2 ∈ k, k þ N ð for which a p n 1 ð Þ ¼ α 1 > α 0 and a p n 2 ð Þ ¼ α 2 > α 0 , then n 1 ¼ m 1 p α 1 , n 2 ¼ m 2 p α 2 hence p α 0 þ1 |n 1 À n 2 . We have p α 0 þ1 < |n 1 À n 2 | ≤ N, what is a contradiction with p α 0 þ1 > N. When we omit such an n 1 from the sum, the error is less than 1

Number Theory and Its Applications
We are going to estimate the first factor of Eq. (3) and simple estimations give Estimate the second factor Eq. (3) 1 N Let N ! ∞, from Eqs. (4) and (5)  Remark. It is known that I u ⊊ I d (see e.g. [5,6]) but the ideals I c and I u are not disjoint, and moreover I u ⊈ I c and I c ⊈ I u . For example the set of all prime numbers belongs to I u but not belongs to I c . On the other hand there exists the set which not belongs to I u but it belongs to I c .
Under the fact that I q  [20]). We have Proof. Let k ∈  and k ≥ 2. It is easy to see that the following equality holds where  denotes the set of all primes. The right-hand side of the equality Eq. (6) equals Then for q > 1 k , the product on the right-hand side of the previous equality converges. Thus, the series on the left-hand side of Eq. (6) converges.  [20]). For q ¼ 1, we obtain I c À lim H n ð Þ log n ¼ 0: Proof. We will show that Every non-negative integer n can be represented as n ¼ ab 2 , where a is a squarefree number. Hence H a ð Þ ¼ 1 and If n ∈ A ε ð Þ then from H n ð Þ ≥ ε Á log n we have It is enough to prove that P n ∈ B n À1 < þ ∞. We have We use the inequality S k ¼ P k j¼1 1 j ≤ 1 þ log k for the harmonic series. Then we have the following inequality Because the P 1 b 2 ¼ π 2 6 < þ ∞, it is enough to prove that the For any n ∈  we have n ¼ p a 1 1 ⋯p a k k ≥ 2 H n ð Þ and from this H n ð Þ ≤ log n log 2 . Therefore We have shown that the sum in Eq. (8) is finite and therefore the sum in Eq. (7) is also finite.
Moreover B ∈ I c and because A ε The situation for sequences ω n ð Þ log log n ∞

Conclusions
It turns out that the study of I -convergence of arithmetical functions or some sequences related to these arithmetical functions for different kinds of ideals I (see [18]) gives a deeper insight into the behaviour and properties of these arithmetical functions.
On the other hand Algebraic number theory has many deep applications in cryptology. Many basic algorithms, which are widely used, have its security due to ANT. The theory of arithmetic functions has many connections to the classical ciphers, and to the general theory as well.