Arithmetical Functions Associated with the -ary Divisors of an Integer

Abstract

The -ary divisibility relations are a class of recursively defined relations beginning with standard divisibility and culminating in the so-called infinitary divisibility relation. We examine the summatory functions corresponding to the -ary analogues of various popular functions in number theory, proving various results about the structure of the -ary divisibility relations along the way.

1. Introduction

Let be a positive integer and denote the set of divisors of by . The set of unitary divisors of , denoted by , are the divisors of which satisfy ; in other words, . The biunitary divisors of are the divisors of which satisfy . This differs from some definitions of biunitary divisibility in the literature (e.g., [1]) but is consistent with others (e.g., [2]). In general, we may define the -ary divisors of an integer to be the set where we define the greatest common -ary divisor of and by We write if .

The -ary divisibility relations as defined above were first introduced by Cohen [3] and have been studied more recently by Haukkanen [2] and Steuding et al. [4]. An alternative definition can be seen in Suryanarayana [5].

One easily verifies the following basic properties:(i) and for all .(ii)If and are coprime, then , where .(iii)If , then .

For example, the set of unitary divisors of a prime power are . On the other hand, the biunitary divisors of a prime power are given by when is odd and when is even. We may then form the unitary and biunitary divisors of a positive integer by “multiplying” the prime-power divisor sets that form the prime decomposition of .

By viewing the sets as representing some of the -convolutions of Narkiewicz [6], we may define the -ary convolution of arithmetic functions and :

The following properties of -ary convolution can be found in [2]:(i)The -ary convolution is commutative.(ii)The function , which takes on value of 1 if and 0 otherwise, is the identity under -ary convolution.(iii)If an arithmetic function satisfies , then possesses a unique inverse under -ary convolution.(iv)If and are multiplicative functions, then is multiplicative as well.

By choosing and appropriately, we may obtain multiplicative -ary analogues to the following classical functions from number theory in terms of the -ary convolution:(i)Let for all . Then is the number of -ary divisors of .(ii)Let and for all . Then is the sum of the -ary divisors of .(iii)Let and for all , where is the unique inverse of the function under the -ary convolution. Then is the analogue of the Euler totient function.(iv)Let and . Then is the analogue of the Dedekind -function.

Note that while counts the totatives of , in general does not count the -ary totatives of .

In this paper, we prove results concerning the structure of -ary divisibility relations and use that to obtain formulae for the number of integers less than or equal to which satisfy . We then apply this result to obtain asymptotics for the summatory functions of -ary generalizations of the classical functions mentioned above.

2. The Behavior of -Ary Divisibility Relations

Let be the set of infinitary divisors of introduced and studied by Cohen [3, 7]. The infinitary divisibility relation can be thought of as the end behavior of the recursion defining the -ary divisibility relations. It satisfies(i)All properties of -ary divisibility relations listed above(ii) if and only if (iii) being transitive; that is, for all , if , then .

Additionally, the following reformulation of Theorem 1 from [3] characterizes in what sense -ary divisibility relations “approach” the infinitary divisibility relation as increases.

Theorem 1. Let be given, and suppose that is such that, for every prime , , where is the exponent of the prime in the prime decomposition of (0 if does not divide ). Then .

Proof. We proceed by induction on . We need to only show the result for prime powers , since by the second property listed above we obtain our theorem by multiplicative construction, akin to the treatment of multiplicative arithmetical functions. Therefore, we may speak of a set prime and consider , with . For , we have and .
Now assume that the result holds up to some . Then consider , with such that . Notice that all divisors of , except itself, satisfy , with . Then, for , we have and, for , we have . So, for not equal to 1 or , . But 1 and are in as well, so , and by the second property of and listed above, we are done.

Additionally, we observe the following.

Theorem 2. For all , , , and .

Proof. We will again use induction. One can immediately verify that and ; only differs from at and , where we have and . Assuming that the theorem holds up to some , observe that for each divisor of the condition implies that on account of for each . Then . But then, by the same reasoning, .
To see that is ordered according to the theorem, consider the statement for all . Using the argument from above and the fact that , we conclude that, for all , and hence our relations hold.

We observe that when looking at the -ary divisors of the powers of a specific prime number, there is always an integer after which, for each , the -ary divisors of will be either or .

Theorem 3. Let be given. Then there is an integer such that, for all and all primes , if is odd, and if is even.

Proof. We note first that, for and , we have that and trivially suffice for bounds. Now assume that such an exists for all . We consider the cases even and odd , respectively.
For even , take and let . Then , since and . For , we have as desired. Since -ary divisions are symmetric, this argument holds for as well. We see then that we may take for even .
For odd , take . Let be such that . Then if . This occurs for all and satisfying . If , then , so . In either case, , so that . We then see that we may take for odd .

Definition 4. We denote by the least for a given .

Our next section concerns itself with -ary analogues of some classical results on summatory functions.

3. Summatory Functions

Let be given and let be an arithmetical function constructed as follows: where is a positive integer and is a function such that . We wish to explore the end behavior of the summatory function of :

We will employ techniques already used in [7, 8] to derive the result for the infinitary and unitary cases, respectively.

Definition 5. Let and . We introduce the following function:

For , this function counts the number of integers that are less than and -ary coprime to . It is known that, for , . The summatory functions for may be broken into the case of even or odd , in accordance with whether or .

Theorem 6. Let be an integer. Then, for even , , with and for odd , with the following:(i) is the number of elements (divisors) in .(ii) is the Möbius function corresponding to : , where is the number of distinct prime factors of , counted without multiplicity.(iii)(iv)

Proof. First note that and are well defined: by Theorem 3, for even , the number of integers satisfying the condition for a given integer must be finite, whereas for odd and for each maximal prime power dividing , the number of integers satisfying must be finite, and hence the product over sums of prime powers -ary-coprime to must be finite. Therefore, the sums are finite. We will prove the result for even first.
Let be even and consider whereis the square-free part of the integer , from [8]. Here we have split each uniquely into a part that has no common divisor with and a part whose prime decomposition uses only the primes of (note that there is no restriction on the prime powers used; e.g., may appear in this decomposition for large enough ).
We proceed: using the fact that the behavior of is known. Pulling out the constants with respect to the sum then immediately gives us our result.
For odd , we proceed in a different manner: We then analyze the term We wish to split into as before. However, this should be done in such a way as to be both unique and useful in dealing with the requirement that . For each divisor of , let , with , , and for each . We invoke the principle of inclusion-exclusion, enabling us to write wherewith being an appropriately indexed set of primes, and we use the fact that, for d = 1, the sum is 0.
This simplifies to and our result follows.

We let be the coefficient appearing in front of the “” term in and let be the function in the error term, so that .

Remark 7. Regarding the function , we may estimate that through the following reasoning: for even , consider (see Definition 4). If , then for at most such (excluding ). If , then for at most 1 such (the case of and , as here ). Thus, for . For , , and so for at most 2 such by the above comments.
For odd , a similar argument gives for at most choices of when , and precisely 2 choices of when ; namely, and . So bounds for all .
Now, for all , and for some , . Thus, since for all , we have that . In particular, there is a least such that, for all , . We will use this in our asymptotic estimates.

We immediately get the following result as a consequence of Theorem 6.

Corollary 8. for .

Proof. The case is trivially true. We prove for each using Stieltjes Integration. Then where the error term from the integral is absorbed by .

Theorem 9. Let . Suppose that an arithmetical function is of the form with and and is , with . Then

Proof. Let , , and be given. Then By our remark above, we may find such that for all , which enables us to estimate where we use the fact that is with . Also, and since is and is bounded, the infinite sums converge, but and , so this is absorbed into our error term and we have our result.

Note that, for each and for all , we may state our error term for the summatory function as , where the multiplicative constant implied by the Big-Oh notation depends only on and . This enables us to achieve roughly the same error as Cohen [7], albeit not as asymptotically strong as . However, as tends to infinity, our error becomes unbounded, and so we cannot achieve Cohen’s result for the infinitary case.

We recall that , the -ary analogue of the Möbius function, is defined recursively via which is extended to all by making multiplicative. We then have the following.

Lemma 10. Let be given. Then there is a constant depending only on such that, for each , .

Proof. By Theorem 3, for each prime and each , there is an that ensures or , depending on the parity of , for all . The values of for are finite, being generated from a finite recursion. For odd with , . For even and even , with ,since and , as is odd. But so . Also, sofor even . Hence, is bounded in absolute value for each —call this bound —and so and we are done.

We will analyze the summatory functions for the -ary analogues of several well-known families of arithmetical functions:(i)The -ary divisor sum functions: (ii)The -ary Jordan totient functions: (iii)The -ary Dedekind functions:

Here denotes a positive integer. By Lemma 10, we may apply Theorem 9 to the Jordan and Dedekind functions of order without issue, since is logarithmic in ; the summatory functions for the divisor sum functions carry no special restriction on aside from it being a positive integer:(i)(ii)(iii)

By Theorems 2 and 3, for each , the sequence (resp., ) is monotonically increasing (resp., decreasing). Both sequences must have the same limit, , which one can identify with the function from Cohen’s manuscript. However, we cannot obtain the function via a limit as tends to infinity of , as the error term grows without bound in . A new approach will likely be needed in order to unify the infinite case with the finite cases.

Data Availability

All data used either was obtained via the cited sources or is derived explicitly from first principles.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to acknowledge the Department of Mathematical Sciences at The University of Texas at Dallas for providing them with funding for this endeavor. They would also like to thank Dr. Paul Stanford for his numerous revisions and for inspiring and teaching them all.