Let n > 1 be an integer with its canonical representation, n = p 1 α 1 p 2 α 2 ⋯ p k α k . Put H n = max α 1 … α k , h n = min α 1 … α k , ω n = k , Ω n = α 1 + ⋯ + α k , f n = ∏ d ∣ n d and f ∗ n = f n 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 = I c q = A ⊂ N : ∑ a ∈ A a − q < + ∞ is an admissible ideal on N such that for q ∈ 0 1 we have I c q ⊊ I d , thus I c q –convergence is stronger than the statistical convergence ( I d –convergence).
- arithmetical functions
- normal order
- binomial coefficients
The notion of statistical convergence was introduced independently by Fast and Schoenberg in [1, 2], and the notion of –convergence introduced by Kostyrko et al. in the paper  coresponds to the natural generalization of statistical convergence (see also  where –convergence is defined by means of filter – the 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 . Many authors deal with average and normal order of the well-known arithmetical functions (see [20, 21, 23, 24, 26, 28, 32, 33] and the monograph  for basic properties of the well-known arithmetical functions). In what follows, we shall strengthen these results from the standpoint of –convergence of sequences, mainly by –convergence and –convergence. On connection with that we can obtain a good information about behaviour and properties of the well-known arithmetical functions by investigating –convergence of these functions or some sequences connected with these functions. Specifically in  by means of –convergence, there is recalled the result that normal order of or respectively is . We managed to completely determine for which the sequences and are –convergent. As consequence of our results, we have that the above sequences are –convergent to , what is equivalent that normal order of or respectively is . Further in , there is proved that the sequence is –convergent to (see also ). We shall extend this result by means of –convergence of the sequence . So we can get a better view of the structure of the set , . We also want to investigate the –convergence of further arithmetical functions.
2. Basic notions
Let be the set of positive integers. Let . If , , we denote by the cardinality of the set . is abbreviated by . We recall the concept of asymthotic, logarithmic and uniform density of the set (see [35, 36, 37, 38]).
Definition 1.1. Let .
Put , where is the characteristic function of the set . Then the numbers and are called the lower and upper asymptotic density of the set , respectively. If there exists , then is said to be the asymptotic density of .
Put , where . Then the numbers and are called the lower and upper logarithmic density of , respectively. Similarly, if there exists , then is said to be the logarithmic density of . Since for and is the Euler constant, can be replaced by in the definition of .
Put and . The following limits , exist (see [17, 37, 39, 40]) and they are called lower and upper uniform density of the set , respectively. If , then we denote it by and it is called the uniform density of . It is clear that for each we have
Let be the canonical representation of the integer . Recall some arithmetical functions, which belong to our interest.
– the number of distinct prime factors of ,
– the number of prime factors of counted with multiplicities ,
– the number of divisors of ,
define h(n) and H(n), put , and for denote
, , where ,
is defined as follows: and if , then is a unique integer satisfying , but i.e., ,
and – were introduced in connection with representation of natural numbers of the form , where are positive integers. Let
be all such representations of given natural number , where . Denote by
It is clear that , because for any there exist representation in the form .
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 is the unifying principle how to express that a subset of is small. We say a set is a small set if . Recall the notion of an ideal of subsets of .
Let . is said to be an ideal in , if is additive (if then ) and hereditary (if and then ). An ideal is said to be non-trivial ideal if and . A non-trivial ideal 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 is a filter if and only if , for each we have and for each and each we have (for definitions see e.g. [4, 41, 42]). Let be a proper ideal in (i.e. ). Then a family of sets is a filter in , so called the associated filter with the ideal .
The following example shows the most commonly used admissible ideals in different areas of mathematics.
The class of all finite subsets of forms an admissible ideal usually denoted by .
Let be a density function on , the set is an admissible ideal. We will use namely the ideals , and related to asymptotic, logarithmic and uniform density, respectively.
A wide class of ideals can be obtained by means of regular non negative matrixes (see ). For , we put for . If exists, then is called –density of (see [3, 44]). Put . Then is a non-trivial ideal and contains both and ideals as a special case. Indeed can be obtained by choosing for , for and by choosing for , for where for .
For the matrix , where for , and otherwise we obtain ideal of Schoenberg (see ), where is Euler function.
Another special case of is the following. Take an arbitrary divergent series , where for and put for , where , and for .
Let be a finitely additive normed measure on a field . Suppose that contains all singletons , . Then the family is an admissible ideal. In the case if is the Buck measure density (see [13, 45]), is an admissible ideal and .
Suppose that is a finitely additive normed measure for . If for there exists , then the set is said to be measurable and is called the measure of . Obviously is a finitely additive measure on some field . The family is a non-trivial ideal. For we can take for instance , or .
Let be a decomposition on (i.e. for ). Assume that are infinite sets (e.g. we can choose for ). Denote the class of all such that intersects only a finite number of . Then is an admissible ideal.
For an the set is an admissible ideal (see ). The ideal is usually denoted by . It is easy to see, that for any , we have
The fact that for any , we have in Eq. (2) is clear. For showing that it suffices to find a set such that and . Put . Since and we have . On the other side for , so we obtain since .
4. – and –convergence
Let us recall notions of statistical convergence, – and –convergence of sequence of real numbers (see ).
Definition 1.3. We say that a sequence is statistically convergent to a number and we write , provided that for each we have , where .
We say that a sequence is –convergent to a number and we write , if for each the set belongs to the ideal .
Let be an admissible ideal on . A sequence of real numbers is said to be –convergent to , if there is a set , such that for we have , where the limit is in the usual sense.
In the definition of usual convergence the set is finite, it means that it is small from point of view of cardinality, . Similarly in the definition of statistical convergence the set has asymptotic density zero, it is small from point of view of density, . The natural generalization of these notions is the following, let be an admissible ideal (e.g. anyone from Example 1.2) then for each we ask whether the set belongs in the ideal . In this way we obtain the notion of the –convergence. For the following use, we note that the concept of –convergence can be extended for such sequences that are not defined for all , but only for “almost” all . This means that instead of a sequence we have , where runs over all positive integers belonging to and .
Remember that –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 ).
If and , then .
If and , then .
The following properties are the most familiar axioms of convergence (see ).
(S) Every constant sequence converges to .
(H) The limit of any convergent sequence is uniquely determined.
(F) If a sequence has the limit , then each of its subsequences has the same limit.
(U) If each subsequence of the sequence has a subsequence which converges to , then converges to .
A natural question arises which above axioms are satisfied for the concept of –convergence.
–convergence satisfies (S), (H) and (U).
If contains an infinite set, then –convergence does not satisfy (F).
Theorem 1.7 (see ) Let be an admissible ideal in . If then .
The following example shows that the converse of Theorem 1.7 is not true.
Example 1.8. Let be an ideal from Example 1.2 f). Define as follows: For we put for . Then obviously . But we show that does not hold.
If then directly from the definition of there exists such that . But then and so we have for infinitely many indices . Therefore cannot be true.
In  was formulated a necessary and sufficient condition for an admissible ideal under which – and –convergence to be equivalent. Recall this condition (AP) that is similar to the condition (APO) in [7, 35].
Definition 1.9 (see also ) An admissible ideal is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets belonging to there exists a countable family of sets such that symmetric difference is finite for and .
Remark. Observe that each from the previous Definition belong to .
Theorem 1.10 (see ) From the statement follows if and only if satisfies the condition (AP).
In  it is proved that – and –convergence are equivalent in provided that from Example 1.2 c) is a non-negative triangular matrix with for . From this we get that , , –convergence coinside with , , –convergence, respectively. On the other hand for further ideals from Example 1.2 e.g. , and , respectively, we have that they do not fulfill the assertion that their –convergence coincides with –convergence. Since these ideals do not fulfill condition (AP) (see [13, 38, 40]).
The following Theorem shows that also for all ideals for the concepts – and –convergence coincide.
Proof. It suffices to prove that for any , and any sequence of real numbers such that for there exists a set such that and .
For any positive integer let and . As , we have , i.e.
Therefore there exists an infinite sequence of integers such that for every
Let . Then
Thus . Put . Now it suffices to prove that . Let . Choose such that . Let . Then belongs to some interval where and does not belong to . Hence belongs to , and then for every , thus .
Corollary 1.12 Ideals for have the property (AP).
It is easy to prove the following lemma.
Lemma 1.13 (see ). If then the statement implies .
5. –convergence of arithmetical functions
We can obtain a good information about behaviour and properties of the well-known arithmetical functions by investigating –convergence of these functions or some sequences connected with these functions. Recall the concept of normal order.
Definition 1.14. The sequence has the normal order if for every and almost all (almost all in the sense of asymptotic density) values we have .
Schinzel and Šalát in  pointed out that one of equivalent definitions to have the normal order is as follows. The sequence has the normal order if and only if . The results concerning the normal order will be formulated using the concept of statistical convergence, which coincides with –convergence. 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  there are studied various kinds of convergence of arithmetical functions which were mentioned at the beginning. The following equalities were proved in the paper  by using the concept of the normal order.
Similarly for the functions and . In  it is proved the following equality:
Let us recall one more result from , there was proved that the sequence is –convergent to . Moreover the sequence is –convergent to for and it is not –convergent for all , as it was shown in . In  it was proved that this sequence is also –convergent.
The following theorem shows that the assertions using the notion instead of , need to use a different technique for their proofs. First of all we recall a new kind of convergence so called the uniformly strong –Cesàro convergence. This convergence is an analog of the notion of strong almost convergence (see ).
Definition 1.15. A sequence is said to be uniformly strong –Cesàro convergent to a number if uniformly in .
The following Theorem shows a connection between uniformly strong –Cesàro convergence and –convergence.
Theorem 1.16 (see ). If is a bounded sequence, then is –convergent to if and only if is uniformly strong –Cesàro convergent to for some , .
The sequence is –convergent to zero i.e. for arbitrary the set has uniform density equal to zero.
Theorem 1.17 (see ). We have .
Proof. The sequence is bounded. Using Theorem 1.16, it is sufficient to show that the sequence is uniformly strong –Cesàro convergent to for . For the reason that all members of are positive, we shall prove that , uniformly in . if . Let . This immediately implies that . Then for all we have with the possible exception of one for which we could have . Assume that there exist two such numbers for which and , then , hence . We have , what is a contradiction with . When we omit such an from the sum, the error is less than . Using the Hölder’s inequality we get
We are going to estimate the first factor of Eq. (3)
Formula , where and simple estimations give .
So we get
Estimate the second factor Eq. (3)
uniformly in .
Remark. It is known that (see e.g. [5, 6]) but the ideals and are not disjoint, and moreover and . For example the set of all prime numbers belongs to but not belongs to . On the other hand there exists the set , where which not belongs to but it belongs to .
Under the fact that for all and Lemma 1.13 it is useful to investigate –convergence of special sequences described in the introduction. Under the Lemma 1.13 it is clear that if there exists the –limit of some sequence for any , then it is equal to the –limit of the same sequence. There are no other options.
Consider the sequences and . In  it was proved that these sequences are dense on and moreover they both are statistically convergent to zero. The same result we have for –convergence, but only for the sequence for all .
Theorem 1.18 (see ). We have
Proof. Let and . 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 , the product on the right-hand side of the previous equality converges. Thus, the series on the left-hand side of Eq. (6) converges.
Let . Put . There exists an for all such that for all and we have (it is sufficient to put ).
From this for all , .
Therefore for all and since the series Eq. (6) converges for all . If for sufficient large then for all .
Corollary 1.19. We have
For the sequence we get the result of different character.
Theorem 1.20 (see ). The sequence is not –convergent for every .
Proof. In the paper  is proved, that the sequence is not –convergent for any . The sequence is also not –convergent to zero. The inequality holds for all and for any prime number . Then we have for all . This implies that the sequence is also not –convergent to zero for every .
Theorem 1.21 (see ). For , we obtain
Proof. We will show that
for any .
Every non-negative integer can be represented as , where is a square-free number. Hence and
If then from we have
It is enough to prove that . We have
We use the inequality for the harmonic series. Then we have the following inequality
Because the , it is enough to prove that the
For any we have and from this . Therefore
Moreover and because we have .
The situation for sequences , is following.
Theorem 1.22 (see ). The sequences and are not –convergent for all .
Proof. We prove this assertion only for . The proof for the sequence is analogous. Let . On the basis of the Theorem 2.2 of  and Lemma 1.13 we can assume that . Take and consider the set
Put , where is a prime number, then and holds for all prime numbers . Therefore the set contains all prime numbers greater than . For these we have: and so . From this . Under the inclusion and according to Lemma 1.13 we have for . This complete the proof.
Further possibility where the results can be strengthened by the way that the statistical convergence in them is replaced by –convergence is the concept of the famous Pascal’s triangle. The -th row of the Pascal’s triangle consists of the numbers . Their sum equals to . Let denote the number of times the positive integer , occurs in the Pascal’s triangle. That is, is the number of binomial coefficient satisfying . From this point of view is the function which maps the set in the set . Let us observe that for every , .
In  it is proved that the average and normal order of the function is . Since the normal order is , we have
Theorem 1.23 (see ).
Proof. The values of the function are positive integers for . Thus for the set is a subset of the set , where . Note that . Therefore is suffices to show that . Evidently this is equivalent with
In  it is shown that . Therefore there exists such that for every , holds. But then
According these inequalities by comparison test of the convergence of the series in Eq. (9) follows.
Now we shall use the concept of –convergence.
Theorem 1.24 (see ). For every , holds and does not hold for any , .
Proof. Let and let have the same meaning as in the proof of Theorem 1.23. Let us examine the series . We write it in the form
In virtue of Lagrange’s mean value theorem we have
Therefore the series Eq. (10) can be written in the form
But , and so
We have already seen, that , (in the proof of Theorem 1.23). Consider that every binomial coefficient , occurs in Pascal’s triangle at least four times . Therefore every number of this form belongs to . Consequently for , the number is greater then or equal to the number of all numbers of the form , not exceeding . But , where is the integer satisfying
From this we get
So we obtain
Now it is clear that if then
Therefore by Eq. (11) we get
But , hence and so
From this owing to Eq. (12) we obtain
Thus , and so for every .
Similar results we can prove for functions and .
Proof. According to Theorem 2.1 of  suppose that the
where . Let and define the set
Put , where is a prime number, then and . Therefore the set contains all prime numbers. Next we have:
Hence and for all .
Proof. According to Theorem 2.2 of  again suppose that the
where . The proof is going similar as in the previous Theorem. Put , , where , are distinct prime numbers. Then , . Hence . Let and define the set
This set contains all numbers of the type , . For we have:
Since the series diverges, we have for all . Therefore and the proof is complete.
There exists a relationship between functions and (where is the number of divisors of ). The following equality holds: , (see ). From this we have
From Theorem 1.25 we have the following statement.
Corollary 1.27. The sequence is not –convergent for all .
The following results concerning the functions and .
In [33, Theorem 3, 5] there are proofs of the following results:
In connection with these results we have investigated the convergence of series for any ,
that we need for –convergence of functions and . The following results are outlined in .
Theorem 1.28. The series
diverges for and converges for .
Let . Put . A simple estimation gives
Clearly for . Therefore
Let . We will use the formula
For a sufficiently large number we have . We can estimate the series on the right-hand side of Eq. (14) with
Since we get
Corollary 1.29. The sequence is
i. –convergent to 1,
ii. –divergent for and –convergent to for .
Let . The set of numbers is a subset of and . From the definition of –convergence Cor. 1.29 i. (Cor. is the abbreviation for Corollary) follows.
Let and denote . When then for the numbers , considering Eq. (13) for holds
Therefore is –divergent. If , then and
The convergence of the series on the right-hand side we proved previously in Theorem 1.28. Therefore is –convergent to if .
Remark. We have .
Theorem 1.30 The series
diverges for and converges for .
Proof. Let . We write the given series in the form
For the -th term of we have
Denote by and consider that . Hence the series converges (diverges) if and only if the series converges (diverges). Since is convergent (divergent) for any so does the series and therefore the series .
Corollary 1.31. The sequence is
–convergent to 1,
–divergent for and –convergent to for .
Proof. Similar to the proof of Corollary 1.29.
Remark. We have .
It turns out that the study of –convergence of arithmetical functions or some sequences related to these arithmetical functions for different kinds of ideals (see ) 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.
This part was partially supported by The Slovak Research and Development Agency under the grant VEGA No. 2/0109/18.