Open access peer-reviewed chapter - ONLINE FIRST

# I–Convergence of Arithmetical Functions

By Vladimír Baláž and Tomáš Visnyai

Submitted: November 25th 2019Reviewed: February 28th 2020Published: May 7th 2020

DOI: 10.5772/intechopen.91932

## Abstract

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).

### Keywords

• sequences
• I–convergence
• arithmetical functions
• normal order
• binomial coefficients

## 1. 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 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 [31]. 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 [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 Icq–convergence and Iu–convergence. On connection with that we can obtain a good information about behaviour and properties of the well-known arithmetical functions by investigating I–convergence of these functions or some sequences connected with these functions. Specifically in [28] by means of Id–convergence, there is recalled the result that normal order of Ωnor ωnrespectively is loglogn. We managed to completely determine for which q01the sequences Ωnloglognand ωnloglognare Icq–convergent. As consequence of our results, we have that the above sequences are Id–convergent to 1, what is equivalent that normal order of Ωnor ωnrespectively is loglogn. Further in [26], there is proved that the sequence logpapnlognis Id–convergent to 0(see also [21]). We shall extend this result by means of Iu–convergence of the sequence logpapnlogn. So we can get a better view of the structure of the set Bε=nN:logpapnlogn<ε, ε>0. We also want to investigate the Icq–convergence of further arithmetical functions.

## 2. Basic notions

Let Nbe the set of positive integers. Let AN. If m,nN, mn, we denote by Amnthe cardinality of the set Amn. A1nis abbreviated by An. We recall the concept of asymthotic, logarithmic and uniform density of the set AN(see [35, 36, 37, 38]).

Definition 1.1. Let AN.

1. Put dnA=Ann=1nk=1nχAk, where χAis the characteristic function of the set A. Then the numbers d¯A=liminfndnAand d¯A=limsupndnAare called the lower and upper asymptotic density of the set A, respectively. If there exists limndnA, then dA=d¯A=d¯Ais said to be the asymptotic density of A.

2. Put δnA=1snk=1nχAkk, where sn=k=1n1k. Then the numbers δ¯A=liminfnδnAand δ¯A=limsupnδnAare called the lower and upper logarithmic density of A, respectively. Similarly, if there exists limnδnA, then δA=δ¯A=δ¯Ais said to be the logarithmic density of A. Since sn=logn+γ+O1kfor nand γis the Euler constant, sncan be replaced by lognin the definition of δnA.

3. Put αs=minn0An+1n+sand αs=maxn0An+1n+s. The following limits u¯A=limsαss, u¯A=limsαssexist (see [17, 37, 39, 40]) and they are called lower and upper uniform density of the set A, respectively. If u¯A=u¯A, then we denote it by uAand it is called the uniform density of A. It is clear that for each ANwe have

Further densities can be found in papers [11, 12].

Let n=p1α1p2α2pkαkbe the canonical representation of the integer nN. Recall some arithmetical functions, which belong to our interest.

1. ωn– the number of distinct prime factors of nωn=k,

2. Ωn– the number of prime factors of ncounted with multiplicities Ωn=α1++αk,

3. dn– the number of divisors of ndn=dn1,

4. define h(n) and H(n), put h1=1, H1=1and for n>1denote

hn=min1jkαj,Hn=max1jkαj,

• fn=dnd, fn=fnn, where n=1,2,,

• apnis defined as follows: ap1=0and if n>0, then apnis a unique integer j0satisfying pjn, but pj+1ni.e., papnn,

• γnand τn– were introduced in connection with representation of natural numbers of the form n=ab, where a,bare positive integers. Let

• n=a1b1=a2b2==aγnbγn

be all such representations of given natural number n, where ai,biN. Denote by

τn=b1+b2++bγn,n>1.

It is clear that γn1, because for any n>1there exist representation in the form n1.

## 3. 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 I2Xis the unifying principle how to express that a subset of XØis small. We say a set AXis a small set if AI. Recall the notion of an ideal Iof subsets of N.

Let I2N. Iis said to be an ideal in N, if Iis additive (if A,BIthen ABI) and hereditary (if AIand BAthen BI). An ideal Iis said to be non-trivial ideal if IØand NI. A non-trivial ideal Iis said to be admissible ideal if it contains all finite subsets of N. The dual notion to the ideal is the notion filter. A non-empty family of sets F2Nis a filter if and only if ØF, for each A,BFwe have ABFand for each AFand each BAwe have BF(for definitions see e.g. [4, 41, 42]). Let Ibe a proper ideal in N(i.e. NI). Then a family of sets FI=BN:thereexistsAIsuchthatB=N\Ais a filter in N, 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.

1. The class of all finite subsets of Nforms an admissible ideal usually denoted by If.

2. Let ϱbe a density function on N, the set Iϱ=AN:ϱA=0is an admissible ideal. We will use namely the ideals Id, Iδand Iurelated to asymptotic, logarithmic and uniform density, respectively.

3. A wide class of ideals Ican be obtained by means of regular non negative matrixes T=tn,kn,kN(see [43]). For AN, we put dTnA=k=1tn,kχAkfor nN. If limndTnA=dTAexists, then dTAis called T–density of A(see [3, 44]). Put IdT=AN:dTA=0. Then IdTis a non-trivial ideal and IdTcontains both Idand Iδideals as a special case. Indeed Idcan be obtained by choosing tn,k=1nfor kn, tn,k=0for k>nand Iδby choosing tn,k=1ksnfor kn, tn,k=0for k>nwhere sn=k=1n1kfor nN.

For the matrix T=tn,kn,kN, where tn,k=φknfor kn, knand tn,k=0otherwise we obtain Iφideal of Schoenberg (see [2]), where φis Euler function.

Another special case of IdTis the following. Take an arbitrary divergent series n=1cn, where cn>0for nNand put tn,k=ckSnfor kn, where Sn=i=1nci, and tn,k=0for k>n.

4. Let μbe a finitely additive normed measure on a field S2N. Suppose that Scontains all singletons n, nN. Then the family Iμ=AN:μA=0is an admissible ideal. In the case if μis the Buck measure density (see [13, 45]), Iμis an admissible ideal and IμId.

5. Suppose that μn:2N01is a finitely additive normed measure for nN. If for ANthere exists μA=limnμnA, then the set Ais said to be measurable and μAis called the measure of A. Obviously μis a finitely additive measure on some field S2N. The family Iμ=AN:μA=0is a non-trivial ideal. For μnwe can take for instance dn, δnor dTn.

6. Let N=j=1Djbe a decomposition on N(i.e. DkDl=Øfor kl). Assume that Djj=12are infinite sets (e.g. we can choose Dj=2j12s1:sNfor j=1,2,). Denote INthe class of all ANsuch that Aintersects only a finite number of Dj. Then INis an admissible ideal.

7. For an q01the set Icq=AN:aAaq<+is an admissible ideal (see [23]). The ideal Ic1=AN:aAa1<+is usually denoted by Ic. It is easy to see, that for any q1,q201, q1<q2we have

IfIcq1Icq2IcIdIδ.E2

The fact IcIdin Eq. (2) follows from the following result. Let ANand aA1a<then dA=0(see [46]) thus if AIcthen AId. The opposite is not true, consider the set of primes P, for which we have dP=0but pP1p=thus PIdbut PIcIcId.

The fact that for any q1,q201, q1<q2we have Icq1Icq2in Eq. (2) is clear. For showing that Icq1Icq2it suffices to find a set H=h1<h2<<hk<Nsuch that k=1hkq1=+and k=1hkq2<+. Put hk=k1q1. Since h1<h2<<hk<and hkq1kwe have k=1hkq1k=1k1=+. On the other side hk>k1q1112k1q1for k2, so we obtain k=1hkq22q2k=1kq2q1<+since q2q1>1.

## 4. I– and 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, I– and I–convergence of sequence of real numbers (see [3]).

Definition 1.3. We say that a sequence xnn=1is statistically convergent to a number LRand we write limstatxn=L, provided that for each ε>0we have dAε=0, where Aε=nN:xnLε.

Definition 1.4.

1. We say that a sequence xnn=1is I–convergent to a number LRand we write Ilimxn=L, if for each ε>0the set Aε=nN:xnLεbelongs to the ideal I.

2. Let Ibe an admissible ideal on N. A sequence xnn=1of real numbers is said to be I–convergent to LR, if there is a set HI, such that for M=N\H=m1<m2<<mk<FIwe have limkxmk=L, where the limit is in the usual sense.

In the definition of usual convergence the set Aεis finite, it means that it is small from point of view of cardinality, AεIf. Similarly in the definition of statistical convergence the set Aεhas asymptotic density zero, it is small from point of view of density, AεId. The natural generalization of these notions is the following, let Ibe an admissible ideal (e.g. anyone from Example 1.2) then for each ε>0we ask whether the set Aεbelongs 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 nN, but only for “almost” all nN. This means that instead of a sequence xnn=1we have xssS, where sruns over all positive integers belonging to SNand SFI.

Remember that I–convergence in Rhas many properties similar to properties of the usual convergence. All notions which are used next we considered in real numbers R. The following theorem can be easily proved.

Theorem 1.5 (Theorem 2.1 from [9]).

1. If Ilimxn=Land Ilimyn=K, then Ilimxn±yn=L±K.

2. If Ilimxn=Land Ilimyn=K, then Ilimxnyn=LK.

The following properties are the most familiar axioms of convergence (see [47]).

(S) Every constant sequence xxxconverges to x.

(H) The limit of any convergent sequence is uniquely determined.

(F) If a sequence xnn=1has the limit L, then each of its subsequences has the same limit.

(U) If each subsequence of the sequence xnn=1has a subsequence which converges to L, then xnn=1converges 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 I2Nbe an admissible ideal.

1. I–convergence satisfies (S), (H) and (U).

2. If Icontains an infinite set, then I–convergence does not satisfy (F).

Theorem 1.7 (see [3]) Let Ibe an admissible ideal in N. If Ilimxn=Lthen Ilimxn=L.

The following example shows that the converse of Theorem 1.7 is not true.

Example 1.8. Let I=INbe an ideal from Example 1.2 f). Define xnn=1as follows: For nDjwe put xn=1jfor j=1,2,. Then obviously Ilimxn=0. But we show that Ilimxn=0does not hold.

If HIthen directly from the definition of Ithere exists pNsuch that HD1D2Dp. But then Dp+1N\H=m1<m2<<mk<FIand so we have xmk=1p+1for infinitely many indices kN. Therefore limkxmk=0cannot be true.

In [3] was formulated a necessary and sufficient condition for an admissible ideal Iunder which I– and 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 I2Nis said to satisfy the condition (AP) if for every countable family of mutually disjoint sets A1A2belonging to Ithere exists a countable family of sets B1B2such that symmetric difference AjΔBjis finite for jNand B=j=1BjI.

Remark. Observe that each Bjfrom the previous Definition belong to I.

Theorem 1.10 (see [14]) From Ilimxn=Lthe statement Ilimxn=Lfollows if and only if Isatisfies the condition (AP).

In [44] it is proved that IdT– and IdT–convergence are equivalent in Rprovided that T=tn,kn,kNfrom Example 1.2 c) is a non-negative triangular matrix with k=1ntn,k=1for nN. From this we get that Id, Iδ, Iφ–convergence coinside with Id, Iδ, Iφ–convergence, respectively. On the other hand for further ideals from Example 1.2 e.g. Iu, INand Iμ, 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 Icqfor q01the concepts I– and I–convergence coincide.

Theorem 1.11 (see, [20, 23]) For any q01the notions Icq– and Icq–convergence are equivalent.

Proof. It suffices to prove that for any Icq, q01and any sequence xnn=1of real numbers such that Icqlimxn=Lfor q01there exists a set M=m1<m2<<mk<Nsuch that N\MIcqand limkxmk=L.

For any positive integer klet εk=12kand Ak=nN:xnL12k. As Icqlimxn=L, we have AkIcq, i.e.

aAkaq<.

Therefore there exists an infinite sequence n1<n2<<nk<of integers such that for every k=1,2,

a>nkaAkaq<12k.

Let H=k=1[(nk,nk+1Ak]. Then

aHaqa>n1aA1aq+a>n2aA2aq++a>nkaAkaq+<12+122++12k+<+.

Thus HIcq. Put M=N\H=m1<m2<<mk<. Now it suffices to prove that limkxmk=L. Let ε>0. Choose k0Nsuch that 12k0<ε. Let mk>nk0. Then mkbelongs to some interval njnj+1where jk0and does not belong to Ajjk0. Hence mkbelongs to N\Aj, and then xmkL<εfor every mk>nk0, thus limkxmk=L.

Corollary 1.12 Ideals Icqfor q01have the property (AP).

It is easy to prove the following lemma.

Lemma 1.13 (see [3]). If I1I2then the statement I1limxn=Limplies I2limxn=L.

## 5. I–convergence of arithmetical functions

We can obtain a good information about behaviour and properties of the well-known 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 xnn=1has the normal order ynn=1if for every ε>0and almost all (almost all in the sense of asymptotic density) values nwe have 1εyn<xn<1+εyn.

Schinzel and Šalát in [28] pointed out that one of equivalent definitions to have the normal order is as follows. The sequence xnn=1has the normal order ynn=1if and only if Idlimxnyn=1. The results concerning the normal order will be formulated using the concept of statistical convergence, which coincides with Id–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 [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.

Idlimωnloglogn=IdlimΩnloglogn=1

and

Idlimhnlogn=IdlimHnlogn=0.

Similarly for the functions fnand fn. In [27] it is proved the following equality:

Idlimloglogfnloglogn=Idlimloglogfnloglogn=1+log2.

Let us recall one more result from [26], there was proved that the sequence logpapnlognn=2is Id–convergent to 0. Moreover the sequence logpapnlognn=2is Icq–convergent to 0for q=1and it is not Icq–convergent for all q01, as it was shown in [21]. In [19] it was proved that this sequence is also Iu–convergent.

The following theorem shows that the assertions using the notion Iuinstead of Icq, q01need 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 [6]).

Definition 1.15. A sequence xnn=1is said to be uniformly strong –Cesàro convergent 0<<to a number Lif limN1Nn=k+1k+NxiL=0uniformly in k.

The following Theorem shows a connection between uniformly strong –Cesàro convergence and Iu–convergence.

Theorem 1.16 (see [6]). If xnn=1is a bounded sequence, then xnn=1is Iu–convergent to Lif and only if xnn=1is uniformly strong –Cesàro convergent to Lfor some , 0<<.

The sequence logpapnlognn=2is Iu–convergent to zero i.e. for arbitrary ε>0the set Aε=nN:logpapnlognε>0has uniform density equal to zero.

Theorem 1.17 (see [19]). We have Iulimlogpapnlogn=0.

Proof. The sequence logpapnlognn=2is bounded. Using Theorem 1.16, it is sufficient to show that the sequence logpapnlognn=2is uniformly strong –Cesàro convergent to 0for =1. For the reason that all members of logpapnlognn=2are positive, we shall prove that limN1Nn=k+1k+Napnlogn=0, uniformly in k. apn=αif pαn. Let α0=logNlogp. This immediately implies that pα0N<pα0+1. Then for all nkk+Nwe have apn=α<α0with the possible exception of one n1kk+Nfor which we could have apn1=α1>α0. Assume that there exist two such numbers n1,n2kk+Nfor which apn1=α1>α0and apn2=α2>α0, then n1=m1pα1, n2=m2pα2hence pα0+1n1n2. We have pα0+1<n1n2N, what is a contradiction with pα0+1>N. When we omit such an n1from the sum, the error is less than 1Napn1logn11Nα1α1logp. Using the Hölder’s inequality we get

1Nn=k+1k+Napnlogn1Nn=k+1k+Napn21Nn=k+1k+N1logn2.E3

We are going to estimate the first factor of Eq. (3)

1Nn=k+1k+Napn21Nα=0α0α2n=k+1apn=αk+N1=1Nα=0α0α2k+Npαkpαk+Npα+1kpα+1=1Nα=0α0α2NpαNpα+1+O1=1NN11pα=0α0α21pα+O1Nα=0α0α2.

Formula α=0α0α2=Pα0, where Px=xx+12x+16and simple estimations give α=0α0α21pαα=0α2pα<.

So we get

1Nn=k+1k+Napn2=O1.E4

Estimate the second factor Eq. (3)

1Nn=k+1k+N1logn21Nn=2N+11logn20,since1logn20.E5

Let N, from Eqs. (4) and (5) we obtain

1Nn=k+1k+Napnlognc1Nn=2N+11logn20,

uniformly in k.

Remark. It is known that IuId(see e.g. [5, 6]) but the ideals Icand Iuare not disjoint, and moreover IuIcand IcIu. For example the set of all prime numbers belongs to Iubut not belongs to Ic. On the other hand there exists the set B=k=1Bk, where Bk=k3+1k3+2k3+kwhich not belongs to Iubut it belongs to Ic.

Under the fact that IcqIdfor all q01and Lemma 1.13 it is useful to investigate Icq–convergence of special sequences described in the introduction. Under the Lemma 1.13 it is clear that if there exists the Icq–limit of some sequence for any q01, then it is equal to the Id–limit of the same sequence. There are no other options.

Consider the sequences hnlognn=2and Hnlognn=2. In [45] it was proved that these sequences are dense on 01log2and moreover they both are statistically convergent to zero. The same result we have for Icq–convergence, but only for the sequence hnlognn=2for all q01.

Theorem 1.18 (see [20]). We have

Icqlimhnlogn=0,forallq01.

Proof. Let kNand k2. It is easy to see that the following equality holds

1+nNhnknq=pP1+1pkq+1pk+1q+,E6

where Pdenotes the set of all primes.

The right-hand side of the equality Eq. (6) equals

pP1+1pkq111pq=pP1+1pk1qpq1.

Then for q>1k, 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 ε>0. Put Aε=nN:hnlognε>0. There exists an n0kNfor all k2such that for all n>n0kand nAεwe have hnεlogn>k(it is sufficient to put n0k=ekε).

From this Aεn0k+1n0k+2nN:hnkfor all k2, kN.

Therefore nAεnq<+for all k2and Icqlimhnlogn=0since the series Eq. (6) converges for all q>1k. If kfor sufficient large then Icqlimhnlogn=0for all q01.

Corollary 1.19. We have

Icqlimhnlogn=0forallq01.

For the sequence Hnlognn=2we get the result of different character.

Theorem 1.20 (see [20]). The sequence Hnlognn=2is not Icq–convergent for every q01.

Proof. In the paper [21] is proved, that the sequence logpapnlognn=2is not Icq–convergent for any q01. The sequence apnlognn=2is also not Icq–convergent to zero. The inequality Hnapnholds for all n=1,2,and for any prime number p. Then we have Hnlognapnlognfor all n=2,3,. This implies that the sequence Hnlognn=2is also not Icq–convergent to zero for every q01.

Theorem 1.21 (see [20]). For q=1, we obtain

IclimHnlogn=0.

Proof. We will show that

Aε=nN:HnlognεIc

for any ε>0.

Every non-negative integer ncan be represented as n=ab2, where ais a square-free number. Hence Ha=1and

HnHb2Hb2+1.

If nAεthen from Hnεlognwe have

logab2Hb2+1εandsologaHb2+1ε.

Therefore

AεB=nN:n=ab2logaHb2+1εbN.

It is enough to prove that nBn1<+. We have

nB1n=b=11b2logaHb2+1ε1a.

We use the inequality Sk=j=1k1j1+logkfor the harmonic series. Then we have the following inequality

nB1nb=11b2Hb2+1ε+1.E7

Because the 1b2=π26<+, it is enough to prove that the

b=1Hb2b2<+.E8

For any nNwe have n=p1a1pkak2Hnand from this Hnlognlog2. Therefore

b=1Hb2b22log2b=1logbb2<+.

We have shown that the sum in Eq. (8) is finite and therefore the sum in Eq. (7) is also finite.

Moreover BIcand because AεBwe have AεIc.

The situation for sequences ωnloglognn=2, Ωnloglognn=2is following.

Theorem 1.22 (see [20]). The sequences ωnloglognn=2and Ωnloglognn=2are not Icq–convergent for all q01.

Proof. We prove this assertion only for ωnloglognn=2. The proof for the sequence Ωnloglognn=2is analogous. Let q=1. On the basis of the Theorem 2.2 of [28] and Lemma 1.13 we can assume that Iclimωnloglogn=1. Take ε012and consider the set

Aε=nN:ωnloglogn1ε.

Put n=p, where pis a prime number, then ωp=1and 1loglogp1εholds for all prime numbers p>p0. Therefore the set Aεcontains all prime numbers greater than p0. For these pwe have: p>p01p=+and so AεIc. From this Iclimωnloglogn1. Under the inclusion IcqIc1Icand according to Lemma 1.13 we have Icqlimωnloglogn1for q01. This complete the proof.

Further possibility where the results can be strengthened by the way that the statistical convergence in them is replaced by Icq–convergence is the concept of the famous Pascal’s triangle. The n-th row of the Pascal’s triangle consists of the numbers n0,n1,,nn1,nn. Their sum equals to 2n=1+1n=k=0nnk. Let Γtdenote the number of times the positive integer t, t>2occurs in the Pascal’s triangle. That is, Γtis the number of binomial coefficient nksatisfying nk=t. From this point of view Γis the function which maps the set Nin the set N0Γ1=0. Let us observe that for every tN, Γt1.

In [32] it is proved that the average and normal order of the function Γis 2. Since the normal order is 2, we have

IdlimΓt=2

(see [28]). We are going to show two results which strengthen the result of [32] and their proofs are outlined in [24].

Theorem 1.23 (see [24]). IclimΓt=2.

Proof. The values of the function Γare positive integers for t1. Thus for ε>0the set Aε=tN:Γt2εis a subset of the set H=12M, where M=tN:Γt>2. Note that Γ2=1. Therefore is suffices to show that nH1n<+. Evidently this is equivalent with

nM1n<+.E9

We shall prove Eq. (9). Firstly, we write the left-hand site of Eq. (9) in the form

nM1n=M11+M2M12++MkMk1k+=M112+M223++Mkkk+1+.

In [32] it is shown that Mx=Ox. Therefore there exists such c1>0that for every kN, Mkc1kholds. But then

Mkkk+1c1k32k=12.

According these inequalities by comparison test of the convergence of the series in Eq. (9) follows.

Now we shall use the concept of Icq–convergence.

Theorem 1.24 (see [24]). For every q>12, IcqlimΓt=2holds and IcqlimΓt=2does not hold for any q, 0<q12.

Proof. Let 0<q<1and let Mhave the same meaning as in the proof of Theorem 1.23. Let us examine the series nM1nq. We write it in the form

nM1nq=M11q+M2M12q++MkMk1kq+=M111q12q++Mk1kq1k+1q+=k=1Mk1kq1k+1q.E10

In virtue of Lagrange’s mean value theorem we have

k+1qkq=qzkq1,k<zk<k+1k=12.

Therefore the series Eq. (10) can be written in the form

kM1kq=k=1Mkqzkq1kqk+1q=k=1qMkkqk+1qzk1q.E11

But zk1q>k1q, k+1q>kqand so

nM1nqqk=1Mkk1+q.

We have already seen, that Mkc1k, k=12(in the proof of Theorem 1.23). Consider that every binomial coefficient t=n2, n4occurs in Pascal’s triangle at least four times asn2nn2t1tt1. Therefore every number of this form belongs to M. Consequently for x>4, xNthe number Mxis greater then or equal to the number Vxof all numbers of the form n2, n4not exceeding x. But Vxs3, where sis the integer satisfying

s2x<s+12.

From this we get

x<ss+12,s>2x1.

So we obtain

MxVxs3>2x4,Mxx>24x21c2>0.

Now it is clear that if x42then

c2xMxc1x,c2=21>0.E12

Therefore by Eq. (11) we get

nM1nq=qk=1Mkkqk+1qzk1q.

But zk<k+1, hence zk1q<k+11qand so

nM1nqqk=1Mkk+11+q.

From this owing to Eq. (12) we obtain

nM1nqqk=1c2kk+11+qqc22k=11k+112+q=+if0<q12.

Thus nM1nq=+, and so nAε1nq=+for every ε>0.

Similar results we can prove for functions fnand fn.

Theorem 1.25 (see [20, 27]). The sequence loglogfnloglognn=2is not Icq–convergent for all q01.

Proof. According to Theorem 2.1 of [27] suppose that the

Icqlimloglogfnloglogn=1+log2,

where q01. Let ε0log2and define the set

Aε=nN:loglogfnloglogn1+log2ε.

Put n=p, where pis a prime number, then fp=pand loglogploglogp=1. Therefore the set Aεcontains all prime numbers. Next we have:

nAεnqj=1pjqj=1pj1=+,q01.

Hence AεIcqand Icqlimloglogfnloglogn1+log2for all q01.

Theorem 1.26 (see [20, 27]). The sequence loglogfnloglognn=2is not Icq–convergent for all q01.

Proof. According to Theorem 2.2 of [27] again suppose that the

Icqlimloglogfnloglogn=1+log2,

where q01. The proof is going similar as in the previous Theorem. Put n=pipj, ij, where pi, pjare distinct prime numbers. Then fn=fpipj=fpipjpipj=pipjpipjpipj=pipj, ij. Hence loglogfpipjloglogpipj=1. Let ε0log2and define the set

Aε=nN:loglogfnloglogn1+log2ε.

This set contains all numbers of the type pipj, ij. For q01we have:

nAεnqj=1pj212pj,pi=2.

Since the series j=112pjdiverges, we have AεIcqfor all q01. Therefore Icqlimloglogfnloglogn1+log2and the proof is complete.

There exists a relationship between functions fnand dn(where dnis the number of divisors of n). The following equality holds: logfn=dn2logn, n>e(see [34]). From this we have

loglogfn=log12+logdn+loglogn,n>ee.

Therefore

loglogfnloglogn=1+logdnloglogn+log12loglogn,n>ee.

From Theorem 1.25 we have the following statement.

Corollary 1.27. The sequence logdnloglognn=2is not Icq–convergent for all q01.

The following results concerning the functions γnand τn.

In [33, Theorem 3, 5] there are proofs of the following results:

n=2γn1n=1,n=2τn1n=1+π26.

In connection with these results we have investigated the convergence of series for any α01,

n=2γn1nα,n=2τn1nα,

that we need for Icq–convergence of functions γnand τn. The following results are outlined in [21].

Theorem 1.28. The series

n=2γn1nα

diverges for 0<α12and converges for α>12.

Proof.

1. Let 0<α12. Put K=k2:kNk>1. A simple estimation gives

n=2γn1nαnKγn1nα.

Clearly γn2for nK. Therefore

n=2γn1nαnK1nα=k=21k2αk=21k=+.E13

1. Let α>12. We will use the formula

n=2γn1nα=k=2s=21kαs=k=21kαkα1.E14

For a sufficiently large number kk>k0we have kαkα1<2. We can estimate the series on the right-hand side of Eq. (14) with

k=21kαkα1<k=2k01kαkα1+2k>k01k2α.

Since 2α>1we get

n=2γn1nα<+.

Corollary 1.29. The sequence γnis

i. Ic–convergent to 1,

ii. Icq–divergent for q012and Icq–convergent to 1for q121.

Proof.

1. Let ε>0. The set of numbers nNn>1:γn1εis a subset of H=n=ts:nNt>1s>1and aH1a<+. From the definition of Ic–convergence Cor. 1.29 i. (Cor. is the abbreviation for Corollary) follows.

2. Let ε>0and denote Aε=nN:γn1ε. When 0<q12then for the numbers nK, K=k2:kNk>1considering Eq. (13) for q=αholds

nAε1nαnK1nα+.

Therefore γnis Icq–divergent. If 12<q<1, q=αthen AεHand

n=21nαk=2s=21kαs.

The convergence of the series on the right-hand side we proved previously in Theorem 1.28. Therefore γnis Icq–convergent to 1if q121.

Remark. We have limstat γn=1.

Theorem 1.30 The series

n=2τn1nα

diverges for 0<α12and converges for α>12.

Proof. Let 0<α<1. We write the given series in the form

n=2τn1nα=k=2s=2skαs,E15

We shall try to use a similar method to Mycielski’s proof of the convergence of n=2τn1nαto explain the equality Eq. (15). Since skαs=kαddt1tαst=kand s=21tαs=1tαtα1the right-hand side of Eq. (15) is equal to

s=22kα1kαkα12=s=2ak

For the k-th term of akwe have

ak=21kα11kα21k2α.

Denote by bk=1k2αand consider that limkakbk=2. Hence the series s=2akconverges (diverges) if and only if the series s=2bkconverges (diverges). Since bkis convergent (divergent) for any α>120<α12so does the series akand therefore the series τn1nα.

Corollary 1.31. The sequence τnis

1. Icconvergent to 1,

2. Icqdivergent for q012and Icq–convergent to 1for q121.

Proof. Similar to the proof of Corollary 1.29.

Remark. We have limstatτn=1.

## 6. 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.

## Acknowledgments

This part was partially supported by The Slovak Research and Development Agency under the grant VEGA No. 2/0109/18.

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Vladimír Baláž and Tomáš Visnyai (May 7th 2020). I–Convergence of Arithmetical Functions [Online First], IntechOpen, DOI: 10.5772/intechopen.91932. Available from: