Arithmetical Functions Associated with the kary Divisors of an Integer

Copyright © 2018 Joseph Vade Burnett et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The k-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 k-ary analogues of various popular functions in number theory, proving various results about the structure of the k-ary divisibility relations along the way.


Introduction
Let  be a positive integer and denote the set of divisors of  by ().The set of unitary divisors of , denoted by  1 (), are the divisors  of  which satisfy () ∩ (/) = {1}; in other words, (, /) = 1.The biunitary divisors of  are the divisors  of  which satisfy  1 ()∩ 1 (/).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 (, )  fl max {  () ∩   ()} . ( 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].
For example, the set of unitary divisors of a prime power   are  1 (  ) = {1,   }.On the other hand, the biunitary divisors of a prime power   are given by (  ) when  is odd and (  ) \ { /2 } 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  = 1 and 0 otherwise, is the identity under -ary convolution.
(iii) If an arithmetic function  satisfies (1) ̸ = 0, then  possesses a unique inverse under -ary convolution.By choosing  and  appropriately, we may obtain multiplicative -ary analogues to the following classical functions from number theory in terms of the -ary convolution: is the number of -ary divisors of .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 (, )  = 1.We then apply this result to obtain asymptotics for the summatory functions of -ary generalizations of the classical functions mentioned above.

The Behavior of 𝑘-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 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  ⩾ 0 be given, and suppose that  ∈ N is such that, for every prime , ]  () ⩽  + 1, where ]  () is the exponent of the prime  in the prime decomposition of  (0 if  does not divide ).Then   () =  ∞ ().
Additionally, we observe the following.
To see that  ∞ () is ordered according to the theorem, consider the statement  ∞ () ⊆ () for all .Using the argument from above and the fact that ) 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  2 (  ) or  1 (  ).Theorem 3. Let  > 2 be given.Then there is an integer   such that, for all  >   and all primes ,   ( Proof.We note first that, for  = 1 and  = 2, we have that  1 = 1 and  2 = 3 trivially suffice for bounds.Now assume that such an   exists for all  = −1.We consider the cases even  and odd , respectively. For even , take  > 2 −1 and let  ⩽ /2.Then   ( − ) ∩   (  ) = {1}, since   ( − ) =  1 ( − ) and  <  − .For 2 = , we have   ∉   (  ) as desired.Since -ary divisions are symmetric, this argument holds for   = − as well.We see then that we may take   = 2 −1 for even .
This occurs for all  and  satisfying  ̸ = 3.If  = 3, then  >  −1 , so   (  ) =  2 (  ).In either case, We then see that we may take   > 3 −1 for odd .Definition 4. We denote by  ⋆  the least   for a given .
International Journal of Mathematics and Mathematical Sciences 3 Our next section concerns itself with -ary analogues of some classical results on summatory functions.

Summatory Functions
Let  > 0 be given and let () be an arithmetical function constructed as follows: where  is a positive integer and () is a function such that () = O( −1 ).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  ⩾ 0 and  > 0. We introduce the following function: For  = 0, this function counts the number of integers  that are less than  and -ary coprime to .It is known that, for  = 0,  0,0 (, ) = (()/) + O(()).The summatory functions for  ,0 may be broken into the case of even  or odd , in accordance with whether   () ⊆  ∞ () or  ∞ () ⊆   ().Theorem 6.Let  > 0 be an integer.Then, for even ,  ,0 (, ) = (()/)  () + O(τ  ()()), with and with the following: is the number of elements (divisors) in  1 ().
(ii)  1 () is the Möbius function corresponding to  1 :  1 () = (−1) () , where () is the number of distinct prime factors of , counted without multiplicity.(iii) Proof.First note that   () and τ () are well defined: by Theorem 3, for even , the number of integers  satisfying the condition (, )  = 1 for a given integer  must be finite, whereas for odd  and for each maximal prime power dividing , the number of integers satisfying (  ,  ]  () )  > 1 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 where core is the square-free part of the integer  2 , 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.,  2 =  2 may appear in this decomposition for large enough ).
We proceed: using the fact that the behavior of  0,0 (, ) is known.Pulling out the constants with respect to the sum then immediately gives us our result.

International Journal of Mathematics and Mathematical Sciences
For odd , we proceed in a different manner: 1. (15) We then analyze the term We wish to split  into  1  2 as before.However, this should be done in such a way as to be both unique and useful in dealing with the requirement that (, )  > 1.For each divisor  of , let  =  1  2 , with core( 2 ) |  | , ( 1 , ) = 1, and ( 2 ,  ]  () )  > 1 for each  | .We invoke the principle of inclusion-exclusion, enabling us to write (  (  (  (  (  (  (  (  (  (  (  (  (  (  (  ( ( where with   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  ,0 (, ) and let   () be the function in the error term, so that  ,0 (, ) =   () + O(()).
We immediately get the following result as a consequence of Theorem 6.
Proof.The case  = 0 is trivially true.We prove for each  > 0 using Stieltjes Integration.Then where the error term from the integral is absorbed by O(    ()).
Theorem 9. Let  ⩾ 0. Suppose that an arithmetical function  is of the form where we use the fact that  is O(  ) with  ⩽  − 1.Also, and since  is O(  ) and   () is bounded, the infinite sums converge, but and  < , so this is absorbed into our error term and we have our result.
International Journal of Mathematics and Mathematical Sciences Note that, for each  and for all  > 0, we may state our error term for the summatory function as O( + ), 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   (log )   .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.
We will analyze the summatory functions for the ary analogues of several well-known families of arithmetical functions: (i) The -ary divisor sum functions: Here  denotes a positive integer.By Lemma 10, we may apply Theorem 9 to the Jordan and Dedekind functions of order  > 0 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: ) 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  ∞,0 (, ) via a limit as  tends to infinity of  ,0 (, ), 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.

2 International
Journal of Mathematics and Mathematical Sciences (iv) If  and  are multiplicative functions, then ⋆   is multiplicative as well.