SOME CHARACTERIZATIONS OF TOTIENTS

. An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler’s phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we )resent several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.


INTRODUCTION
The Dirichlet eonvolut;on of arithmetical functions f and g is defined by If I(1) # 0, then the Dirichlet inverse of f is the arithmetical function f-1 satisfying f * f-f-* f o, where e0(1) 1 and e0(r) 0 for n > 1.An arithmetical function is said to be multiplicative if f(1) 1 and f(mn) f(m)f(n) whenever (rn, n) 1.A multiplicative function f is totally determined by its values f(pe) at all prime powers pe.A multiplicetive function f is also totally determined by its generating series at all primes p.A multiplicative function f is said o be completely multiplicative if f(rnn) f(m.)f(n) for all rn and n.A completely multiplicative function f is totally determined by its values f(p) at all primes p.
In this paper we consider multiplicative functions f which can be written in the form where ft and f, are completely multiplicative functions.Such functions f are called totients ( [14], p. 612).The functions ft and f,, are said to be integral and inverse components of f, respectively.Note that ft and fv are not always unique.Namely, if there exists a prime p such that .f(pe) 0for all positive integers e, then it suffices that ft(p) and fv(p) satisfy ft(p) ft,(p).Otherwise ft(p) and f,(p) are uniquely determined.
Totients belong to the class of rational arithmetical functions.Namely, Vaidyanathaswamy [14] defined an arithmetical function f to be a rational arithmetical function of degree (r,s) if there exist completely multiplicative functions gl,g2,... ,g,.and hl, h2,... ,h, such that Totients are thus rational arithmetical functions of degree (1, 1), and completely multiplicative functions are rational arithmetical functions of degree (1, 0).Note that all completely multi- plicative functions are totients with f. e0.
There is a large number of characterizations of completely multiplicative functions in the literature.For example, each of the following four conditions is a necessary and sufficient condition for a multiplicative function to be completely multiplicative.
(i) f-l(pe) 0 for all primes p and integers e > 2, (ii) f(pe) f(p)e for all primes p and integers e >_ 1, (iii) f-#f, where is the Mhbius function, (iv) fp(z)= for all primes p.
For further characterizations of completely multiplicative functions, see e.g.[I].
Characterizations of totients have not much been studied in the literature.Wall and Hsu [15] have given a characterization.Namely, a mu]tip]icative function f is a totient if and only if f(p), f(p2), f(p3),.., is a geometric progression for each prime p.The purpose of this paper is to present some further characterizations.Most of our characterizations have been developed from known properties of some concrete examples of totients.
Euler's function (n) is a famous concrete totient, which is defined as the number of integers x (mod n) such that (x,n) 1.It is well-known that g , where g(n) n for all n.
Thus is a totient with Ct N and b, e, where e 1.There is a large number of generalizations and analogues of Euler's totient in the literature.Many of the generalizations and analogues are defined combinatorially.For example, the Jordan totient J.(n) is defined as the number of ordered k-tuples (Xl,...,x.) (mod n) such that gcd(xl,x2,...,x.,n) 1.
Many of the generalizations and analogues also possess the structure of a totient in the sense of Vaidyanathaswamy [14].For example, the Jordan totient can be written as Jk N p, where N'(n) n for all n.For further totients reference is made to the books by McCarthy [9] and Sivaramakrishnan [11] and to the paper by Sivaramakrishnan [12].In Section 3 of this paper we introduce a new example of a totient denoted by ().The characterizations of Section 2 can be used in deciding whether a function is a totient and to obtain identities for concrete totients.
We apply the results of Section 2 to the function #(k). 2. CHARACTERIZATIONS In this section we present necessary and sufficient conditions for an arithmetical function f to be a totient.These conditions may not be considered entirely new, since they are either con- cei;ed from properties of some concrete examples totients or known to be necessary conditions for totients.For brevity, we assume throughout, this section that f(1) 1.
THEOREM 1.An arithmetical fimction f is a totient if, and only if, f is multiplicative and for each prime p there exists a complex number a(p) such that f(pe)_ f(p)(a(p))e-1 for all e __> 1. (1) In this case ft(P) a(p).
PROOF.Theorem follows after some manipulation by definition of totients and proper- ties of completely multiplicative functions.We omit the details.COROLLARY 1.If f and g are totients, then fg is a totient with (fg)t ftgt.COROLLARY 2 ([15]).A multiplicative function f is a totient if, and only if, for each prime p, f(p), f(pg.), f(pa) is a geometric progression.
REMARK.Several further characterizations could be derived from Theorem using the formula f(p) ft(p)-f(p) for all primes p. THEOREM 2. An arithmetical function f is a totient if, and only if, there exists com- pletely multiplicative functions g and h such that s() In this case ft =g and fv h.
REMARK.For the Euler totient 4, equation (2 which, in fact, is the definition of .
PROOF.If f is a totient, then We thus arrive at (2).The converse part follows similarly.TttEOREM 3.An arithmetical function f is a totient if, and only if, there exists a multiplicative function h such that al(m,n) In this case f h.REMARK.Sivaramakrishnan ([10], p. 27) and Sugunamma ([13], p. 40) have noted that (3) is a necessary condition for totients.
PROOF.Formula (3) can be proved by showing that both sides of (3) are multiplicative functions in two variables and by studying prime powers.Conversely, taking (m, n) 1 in (3) shows that f is multiplicative.Taking m pe-, n p (e > 9.) in (3) gives Thus, by Theorem 1, we have the result.(4) In this case f, h.
PROOF.Theorem 4 can be proved in a similar way to Theorem 3.
PROOF.Formula ( 5) can be proved by showing that both sides of (5) are multiplicative functions in two variables and by studying prime powers.The converse part follows by taking n in (5).DEFINITION.A divisor d of n is said to be a unitary divisor (or a block divisor) of n if (d, n/d) 1.The greatest common unita divisor of m and n is denoted by (m, n)u.THEOREM 6.If f is a totient, then whenever (m, n)u 1. ConverselF ff here eiss a muliplicive :unction F such f(mn 1() /() + F(p) :o p,im .
REMARK.Identity ( 6) is termed as the BuschRamanujan identity in the literature.Identity (6') with the restriction (m, n)u 1 is called the restricted BuschRamanujan identity.It is well-known that eve totient satisfies the restricted BuschRamanujan identity, s e.g.
[9], p. 53, [14], p. 655.The converse part follows aer laborious elementary computations.We do not present the details here.The converse pa has bn studied in more detail in [6], 3.2.THEOREM 7.An arithmeticM &nction f is a totient with f(p) 0 for prim p i and oy there est a completely multiplicative function g with g(p) 0 d a complex number b(p) for MI prim p such that :(.) #() (1 (p)  () l-I () .q()1-I (),(s') where g is completely nultiplicative with g(p) 0 for all primes p, then f is a totient with ft g" DEFINITION.We define 7 to be the multiplicative function such that 7(pa) p for all primes p. THEOREM 9.If f is a totient, then whenever 7(m) 7(n).Conversely, if f is multiplicative and if there e.xists a completely multiplicative function g such that g(p) 0 for all primes p and g(n)f(m) g(m)f(n), (9') whenever /(m) /(n), then f is a totient with ft g.
Theorems 7-9 followfrom the definition of totients and properties of completely multiplica- tire functions.Details of proofs are left to the reader.DEFINITION.We say that an arithmetical function f satisfies property O if f(p) 0 for a prime p implies f(pe) 0 for all e > 1.
THEOREM 10.An arithmetical function f is a totient if, and only if, f satisfies property 0 and there exists a completely multiplicative function g such that (10) In this case f g.
REMARK.For material relating to the functional equation ( 10) we refer to [3], [4] and PROOF.Formula (10) can easily be seen t(, be valid when m and n are prime powers.Since, in addition, both sides of (10) are multiplicative functions in m and n, we have (10).The converse part follows by taking (m, n) 1 and m p-l, n p (e _> 2) in (10) and applying Theorem 1. COROLLARY 3. If f is an integer-valued totient, then alb f (a) f (b).
PROOF.Since a[b, we have b ac.If c b, then a 1 and the result is true.Suppose that c < b.By (10) we have f(b)f((a,c))--f(ac)f((a,c))= f(a)f(c)ft((a,c)).
If f((a,c)) O, then we can conclude that f(b) 0 and the corollary is valid.Suppose that f ((a, c) -O.Then f(b)-f(a)f(c)ft((a,c))/f((a,c)).Now we can conclude the result inductively.
DEFINITION.Let (m, n)k denote the greatest divisor of n which is a kth power divisor Of ,.
THEOREM 11.Let f and g be arithmetical functions and let f(n) 0 for all n.If I and g are totients with f gt, then (f a)(.
, where (m, nk)k, (11) for all m and n.Conversely, if there exist completely multiplicative functions a, and such that (a f-l)(n (,,d)=t (11') for all n and for some m pe (0 <_ e _< k), then f is a totient with ft .
Assume that (11) holds.We shall firstly prove that f is multiplicative.We proceed by induction on rs to prove that f(rs) f(r)f(s) whenever (r,s) 1.This is valid for rs 1.  Assume that f(ab) f(a)f(b) when (a,b)-1, ab < rs.We distinguish three cases.
Case 2. Let pk m" As rn pe, 0 _ e _( k, we have e k.Thus taking n p in (11') gives f(pa) a(pa-)f(p).By Theorem 1 we now obtain the result.THEOREM 12. Let f and g be arithmetical functions such that (f g)(n) 0 for all n.If f and g are totients with f gt, then f(n) f( 5) - (f --id) .(d),where (m, n), (12) for all m and n.Conversely, if if there exist completely multiplicative functions a, and '1 such that (a fi-1)(n) 0 for all n and such that f(n) (12') for all n and for m pe (0 <_ e <_ k), then f is a tot;ent with ,f: e Theorem 12 can be proved in a similar way to Theorem 1t.We omit the details.
REMARK.Identities (11) and ( 12) can be considered as generalized Landau identities.The classical Landau identity [8] is The classical Brauer-Rademacher identity (see e.g. [9],p. 75)is This identity could also be used as a source of characterizations of totients in a similar way to the Landau identity.We do not present details.The Brauer-Rademacher and the Landau type identities have been considered as characterizations of totients in a very general setting in [7].THEOREM 13.An arithmetical function f is a totient if, and only if, f is multiplicative and if for all primes p there exists complex numbers a(p) and b(p In this case ft(P) a(p) and f,(p) b(p).PROOF.If f is a totient, then (f, X.( Conversely, let g and h be completely multiplicative functions defined by g(p) a(p) and h(p) b(p).Then f,(x) (g h-1)p(x) at all primes p; hence f g h -1.This completes the proof.Theorems 1-12 could easily be generalized to the setting of Narkiewicz's regular convolution (see [9], Chapter 4).We do not present the details here.

AN EXAMPLE
The arithmetical function O(n) is defined as the number of ordered pairs (a, b> such that (a,b) 1 and ab n.It is easy to see that O(n) 2"(n), where w(n) is the number of distinct prime factors of n with w(1) 0. Also O(n) is the number of squarefree divisors of n.This suggests the standard generalization Ok of 0. In fact, Ok(n) is the number of k-free divisors of n (see e.g. [9,p. 36, 37]).In particular, 02 0. The definition of O(n) however suggests we look upon another generalization of O(n).We define O(k)(n) to be the number of ordered k-tuples (al, a2,..., ak) such that a,'s are relatively prime and aa2"'ak n.It is easy to see that O()(n)k (').
The generating series of 0(k) is (O(k)),() a: (14) at all primes p.If we define the function A(1-k) by A(l_k)(n) (1 k) ('), where fl(n) is the total number of prime divisors of n each counted according to its multiplicity, then by Theorem clearly completely multiplicative; hence 0() possesses the structure of a totient.Application of formulas ( 1)-( 12) to 0() yields the following identities.
Formulas (1)-( 13) could also be applied to other concrete totients to obtain identities for these totients.We leave the details to the reader.
Note that 0(k) could also be presented in terms of the unitary convolution.The unitary convolution f @ g of two arithmetical functions f and g is defined by ( g)() ()9(/e), where d[]n means that d is a unitary divisor of n, that is, a divisor d of n with (d, n/d) 1.It is easy to see that O(k) e @ e O e (e, k times).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

THEOREM 4 .
An arithmetical function f is a totient if, and only if, there exists a completely multiplicative function h such that y(mn)y(m) y(n/a)h(a).

THEOREM 8 .
If f is a totient, *hen ]n pin Conversely, if for all n

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