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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 648165, 15 pages
http://dx.doi.org/10.1155/2010/648165
Research Article

Combinatorial Aspects of the Generalized Euler's Totient

1Department of Mathematics, University of the Thai Chamber of Commerce, Bangkok 10400, Thailand
2Department of Mathematics, Faculty of Science and Institute for Advanced Studies, Kasetsart University, Bangkok 10900, Thailand

Received 24 April 2010; Revised 29 July 2010; Accepted 6 August 2010

Academic Editor: Pentti Haukkanen

Copyright © 2010 Nittiya Pabhapote and Vichian Laohakosol. 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.

Abstract

A generalized Euler's totient is defined as a Dirichlet convolution of a power function and a product of the Souriau-Hsu-Möbius function with a completely multiplicative function. Two combinatorial aspects of the generalized Euler's totient, namely, its connections to other totients and its relations with counting formulae, are investigated.

1. Introduction

Let be the unique factorization domain of arithmetic functions [1, 2] equipped with addition and (Dirichlet) convolution defined, respectively, by The convolution identity is defined by For , write for its convolution inverse whenever it exists. A nonzero arithmetic function is said to be multiplicative if and is called completely multiplicative if this equality holds for all . For , the Souriau-Hsu-Möbius (SHM) function ([3, 4], [5, page 107]) is defined by where denotes the unique prime factorization of being the largest exponent of the prime that divides . This function generalizes the usual Möbius function, , because . Note that and for , we have It is easily checked that is multiplicative; there are exactly two SHM functions that are completely multiplicative, namely, and , and there is exactly one SHM function whose convolution inverse is completely multiplicative, namely, . For a general reference on the Möbius function and its generalizations, see Chapter 2 of the encyclopedic work [5].

The classical Euler's totient is defined as the number of positive integers such that . It is well known (page 7 of [1]) that For a general reference about Eulier's totient, its many facets and generalizations, see [5, Chapter 3]. Euler's totient has been given a good deal of generalizations. Of interest to us here is the one due to Wang and Hsu [6], defined for and completely multiplicative by where . In [6] it is shown that possesses properties extending those of the classical Euler totient, such as the following. (P1) when is -powerful, that is, for each prime factor of . (P2)Let (Theorem of [6]). Then, for prime , there uniquely exists an matrix over such that . Let be a subset of . Then, there uniquely exists a completely multiplicative with being defined by the number of vectors in . For , we write , if no row of is in for every prime divisor of . Then for being -powerful, counts the number of -vectors such that .

We take off from the work of Wang and Hsu, by defining our generalized Euler totient (or GET for short) as where , and is a completely multiplicative function. Comparing with the terminology of Wang-Hsu, we see that . For brevity write There have appeared quite a number of results related to our GET, such as those in [4, 69], and the most complete collection to date can be found in [5, Chapter 3]. In the present paper, we consider two aspects of the GET. In the next section, its relations with other totients are investigated. Here we deal mostly with those results closely connected to our GET; for further and more complete collection up to 2004, we refer to the encyclopedic work in [5, Chapter 3]. In the last section, after proving a general inversion formula, various counting formulae related to the GET are derived.

Before listing a few properties of our GET generalizing the classical Euler's totient, we recall some auxiliary notions. The log-derivation, [10], is the operator defined by For , the Rearick logarithmic operator of (or logarithm of ; [1113]), denoted by , is defined via where denotes the log-derivation. For , the Rearick exponential Exp is defined as the unique element such that . For and the th power function is defined as It is not difficult to check that this agrees with the usual power function, should be integral. From [11], we know that if is multiplicative and , then is also multiplicative; the fact which automatically implies its converse.

Proposition 1.1. Let , and be a completely multiplicative function. (A)We have the product representation (B)If , then (C)If , then where

Proof. Part (A) follows immediately from being multiplicative. Part (B) follows from the fact that [14]. To prove Part (C), let where Then Considering (1.16) as a system of simultaneous equations in the unknowns and appealing to Cramer's rule, the result follows.

Part (C) generalizes a well-known identity on page 86 of [2], which is the case where , stating that where

2. Connections with Other Totients

Case I (). When the parameters and take integer values, the GET does indeed represent a number of well-known arithmetic functions, namely, When and , this particular totient is equivalent to quite a few classical totients. (i.1)The Jordan totient which counts the number of -tuples such that and ([1, page 13], [5, pages 186-187, page 275], [2, page 91]). Clearly, . From [5, pages 186-187], closely resembles the Jordan totient is the function While one has, on the other hand, , showing that is not of the form of our GET. Even more general is the Shonhiwa's totient, [5, pages 187, 276], defined as the number of -tuple such that and , whose representation is .(i.2)The von Sterneck function, [1, pages 14-15] and [5, pages 275-276], where the sum is over all ordered -tuples such that and lcm.(i.3)Eckford Cohen's totient which counts the number of elements of a -reduced residue system (mod ). For integers not both , let If , we say that and are relatively -prime. We refer to the subset of a complete residue system (mod ) consisting of all elements of that are relatively -prime to as a -reduced residue system (mod ), [2, pages 98-99] and [5, pages 275-276]. (i.4), where is the Klee's totient, [15], which counts the number of integers for which is th-power-free, that is, contains no th-power divisors other than . The Klee's totient has a product representation of the form, [5, page 278], . (i.5)Haukkanen's totient, [5, page 276], which counts the number of -tuples mod such that .
On the other hand, based on the combinatorial interpretation (P2) of above, our GET includes several special totients [6] such as taking and
(i) when , or when , we obtain Schemmel's totient, , which counts the number of sets of consecutive integers each less than and relatively prime to . The function has a product representation of the form , [5, page 276]. The case was also called Schemmel totient function and was shown by Lehmer to have application in the enumeration of certain magic squares, [5, page 184]. There are many other totients closely connected to Schemmel's totient. As examples, we describe two more, taken from [5, Chapter 3], namely, Lucas's and Nageswara Rao's totients. For fixed integers , Lucas's totient counts the number of integers such that are relatively prime to and its product representation is , where is the number of distinct residues of mod . Nageswara Rao's totient counts the number of sets of consecutive integers each less than which are -prime to ;(ii)let be a set of polynomials with integer coefficients and , we obtain Steven's totient which denotes the number of -vectors such that . Following [5, pages 279-280], the product representation of Stevens's totient takes the form where is the number of incongruent solutions of The Stevens's totient is multiplicative, and contains, as special cases ()the Jordan totient (by taking ); ()the Schemmel totient (by taking ); ()Cashwell-Everett's totient (by taking ), which counts the number of -tuples with such that . Its product representation is
In passing, let us mention that, our GET is closely connected to the generalized Ramanujan sum through

Case II (). The GET also includes a number of known totients in this case. (ii.1)The Garcia-Ligh totient [16, 17], defined for fixed , by where is easily shown to be completely multiplicative. This totient counts the number of elements in the set that are relatively prime to with . (ii.2)The Garcia-Ligh totient is a special case of the following totient taken from Exercise on pages 34-35 of [1]. Let . The number of integers and is, using our terminology above, where is the completely multiplicative function defined over prime by , the number of solutions of the congruence (mod ). (ii.3)Martin G. Beumer's function (Section on pages 72–74 of [2]) defined for , by where we have used a result of Haukkanen [14], that if is a completely multiplicative function and , then . (ii.4)The Dedekind -function ([2, Problem , page 80], [5, page 284]) defined by is clearly equivalent to where is the completely multiplicative function defined for prime by .(ii.5)H. L. Adler's totient ([2, Section , page 102], [5, page 279]) is defined, for fixed , as where is the completely multiplicative function defined, for prime , by The totient value is the number of ordered pairs for which and . Note that when , this is merely Euler's totient. (ii.6)D. L. Goldsmith's totient (the main theorem on page 183 of [18]) is defined, for , by where . From the meaning of , it is clear that is a multiplicative function (Problem , page 31 in [19]). Thus, where is the number of integers such that is divisible by the prime and is the completely multiplicative function defined for prime by . It is possible to enlarge the values of in the Goldsmith's totient such as taking to get where is the completely multiplicative function defined for prime by

Let . For , define the -convolution of and by It is easily checked that the -convolution is neither commutative nor associative. Yet it preserves multiplicativity, that is, if and are multiplicative functions, then the is also multiplicative (Problem , page 37 of [1]). The convolute (page 53 of [1]) of is defined by The -convolution is connected to the usual (Dirichlet) convolution via We list here some examples of arithmetic functions which enjoy -convolution relations. ()Klee's totient, , which counts the number of integers for which is -power-free, satisfies (Problem on pages 38-39 of [1]), ()The number of divisors function is related to the arithmetic function which counts the number of -free divisors of by (Problem , page 37 of [1]) ()The Liouville's function is defined by It is known (Problem on page 45 of [1]) that is completely multiplicative and satisfies ()For , the Gegenbauer's function (page 55 of [1]) is defined as

The notion of convolute enables us to give swift proofs of a number of identities such as the following ones which are generalizations of Problem on page 48, Problem on page 51, Problem on page 55, and Problem on page 56 of [1].

Proposition 2.1. Let . Then, (i), (ii),(iii),(iv),(v)

Proof. We first note the identity which follows from the facts that is multiplicative and Assertion (i) follows quickly from
Assertion (ii) follows from
Assertion (iii) follows from
Assertion (iv) follows from To prove (v), we need the identity . Now Assertion (v) follows from

3. Inversion and Counting Formulae

In [20], see also Problem , pages 36-37 of [1], Suryanarayana proved the following inversion formula:

Our objective now is to extend this inversion formula using our GET.

Theorem 3.1 (modified generalized Möbius inversion formula). Let and . For , one has

Proof. Recall that the summation over , with fixed , means that is written as when runs through all divisors of for which can be so written with . The result follows at once from

Theorem of [20] is a special case of Theorem 3.1 when . As is well known, the Möbius inversion formula has extensive applications which is also the case of our new inversion formula. Applying Suryanarayana's inversion formula to three examples in the last section, we obtain As an application to counting formulae, taking in Theorem 3.1, we get the following special case of Theorem in [21].

Corollary 3.2. Let and Klee's function be the number of integers such that and is not divisible by the th power of any prime. Then,

Following [5, page 136], a generalization of Klee's function introduced by D. Suryanarayana is the function ; note that

To illustrate another application, consider a function of the form which has been a subject of many investigations such as those in [2225]. This function leads to a formula for the number of primitive elements over a finite field. Let and let all distinct prime factors of be . Define It is known ([24, pages 84–86], [5, pages 191–193]) that a result which generalizes Fermat's little theorem when is prime. This congruence has been much extended in [22, 25]. Dickson [24, pages 84–86] shows a connection with Euler's totient via the identity where the summation runs through proper divisors of ; by a proper divisor of we mean a divisor of that does not divide if .

We aim now to extend these results even further to our GET. For and , define where

Theorem 3.3. Let and with . Then,

Proof. Let By Theorem 3.1, we have On the other hand, from the definitions of and , we get Consequently,

Specializing in Theorem 3.3, we recover the result of Dickson mentioned above, namely, If we take , a prime power, in (3.7), then it is well known [26, page 93] that the number of monic irreducible polynomials of order over GF is and is the number of primitive elements of GF/GP. Taking in Theorem 3.3 yields another beautiful formula

Combining the results of Theorems 3.1 and 3.3 yields another relation between and .

Corollary 3.4. Let and with . Then, where

Acknowledgments

V. Laohakosol was supported by the Commission on Higher Education, the Thailand Research Fund RTA5180005, and KU Institute for Advanced Studies. N. Pabhapote was supported by the University of the Thai Chamber of Commerce.

References

  1. P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
  2. R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, vol. 126 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989.
  3. J.-M. Souriau, “Généralisation de certaines formules arithmétiques d'inversion. Applications,” Revue Sci, vol. 82, pp. 204–211, 1944. View at Google Scholar · View at Zentralblatt MATH
  4. T. C. Brown, L. C. Hsu, J. Wang, and P. J.-S. Shiue, “On a certain kind of generalized number-theoretical Möbius function,” The Mathematical Scientist, vol. 25, no. 2, pp. 72–77, 2000. View at Google Scholar · View at Zentralblatt MATH
  5. J. Sándor and B. Crstici, Handbook of Number Theory. II, Kluwer Academic, Dordrecht, The Netherlands, 2004.
  6. J. Wang and L. C. Hsu, “On certain generalized Euler-type totients and Möbius-type functions,” Dalian University of Technology, China, preprint.
  7. P. Haukkanen, “A further combinatorial number-theoretic extension of Euler's totient,” Journal of Mathematical Research and Exposition, vol. 17, no. 4, pp. 519–523, 1997. View at Google Scholar · View at Zentralblatt MATH
  8. V. Laohakosol and N. Pabhapote, “Properties of rational arithmetic functions,” International Journal of Mathematics and Mathematical Sciences, no. 24, pp. 3997–4017, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. V. Laohakosol, P. Ruengsinsub, and N. Pabhapote, “Ramanujan sums via generalized Möbius functions and applications,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 60528, 35 pages, 2006. View at Google Scholar · View at Zentralblatt MATH
  10. H. N. Shapiro, Introduction to the Theory of Numbers, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1983.
  11. D. Rearick, “Operators on algebras of arithmetic functions,” Duke Mathematical Journal, vol. 35, pp. 761–766, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. D. Rearick, “The trigonometry of numbers,” Duke Mathematical Journal, vol. 35, pp. 767–776, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. V. Laohakosol, N. Pabhapote, and W. Wechwiriyakul, “Logarithmic operators and characterizations of completely multiplicative functions,” Southeast Asian Bulletin of Mathematics, vol. 25, no. 2, pp. 273–281, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. P. Haukkanen, “On the real powers of completely multiplicative arithmetical functions,” Nieuw Archief voor Wiskunde, vol. 15, no. 1-2, pp. 73–77, 1997. View at Google Scholar · View at Zentralblatt MATH
  15. V. L. Klee, Jr., “A generalization of Euler's ϕ-function,” The American Mathematical Monthly, vol. 55, pp. 358–359, 1948. View at Google Scholar
  16. P. G. Garcia and Steve Ligh, “A generalization of Euler's ϕ function,” The Fibonacci Quarterly, vol. 21, no. 1, pp. 26–28, 1983. View at Google Scholar
  17. L. Tóth, “A note a generalization of Euler's ϕ function,” The Fibonacci Quarterly, vol. 25, pp. 241–244, 1982. View at Google Scholar
  18. D. L. Goldsmith, “A remark about Euler's function,” The American Mathematical Monthly, vol. 76, no. 2, pp. 182–184, 1969. View at Publisher · View at Google Scholar
  19. W. J. LeVeque, Topics in Number Theory. Vols. 1 and 2, Addison-Wesley, Reading, Mass, USA, 1956.
  20. D. Suryanarayana, “New inversion properties of μ and μ,” Elemente der Mathematik, vol. 26, pp. 136–138, 1971. View at Google Scholar · View at Zentralblatt MATH
  21. P. J. McCarthy, “On a certain family of arithmetic functions,” The American Mathematical Monthly, vol. 65, pp. 586–590, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. L. Carlitz, “Note on an arithmetic function,” The American Mathematical Monthly, vol. 59, pp. 386–387, 1952. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. L. E. Dickson, “A generalization of Fermat's Theorem,” in Collected Mathematical Papers, vol. 2, Chelsea Publishing, New York, NY, USA, 1975. View at Google Scholar
  24. L. E. Dickson, “Generalizations of Fermat's Theorem,” in History of the Theory of Numbers, vol. 1, Chelsea Publishing, New York, NY, USA, 1999. View at Google Scholar
  25. P. J. McCarthy, “On an arithmetic function,” Monatshefte für Mathematik, vol. 63, pp. 228–230, 1959. View at Google Scholar · View at Zentralblatt MATH
  26. R. Lidl and H. Niederreiter, Finite Fields, vol. 20 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, Mass, USA, 1983.