AN IDENTITY FOR A CLASS OF ARITHMETICAL FUNCTIONS OF TWO VARIABLES

For a positive integer r, let r∗ denote the quotient of r by its largest squarefree divisor (1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p


INTRODUCTION.
For a positive integer r, let r, denote the quotient of r when divided by its largest square free divisor (i, i).Recently K. R. Johnson [i]  P II, r, p+ r according as r, ln or not, where C(n,r) is the well known Ramanujan's sum and pa In means that paln and pa+l # n.In his proof, since C(n,r) is not multiplicative in n, he used [i] two identities concerning C(n,r) that were proved by him in an earlier paper [2].As a matter of fact (i.i) can be obtained directly from the well known pro- perty of the Ramanujan's sum, namely its multiplicativity in both variables n and r.
In section 2, using this method, we generalize (i.i) (see theorem) to a class of arith- metical functions of two variables and in section 3, specializing our theorem, we deduce as corollaries, formulae analogous to (i.I) for several generalizations of the Ramanujan's sum and (i.i) also.
We recall that an arithmetical function f(n,r) is said to be multiplicative in both variables n and r if and that such a function is completely determined by its values f(pa-,Pb) for primes p and non negative integers a and b.
For given arithmetical functions g(n) and h(n) and for a given positive integer k, g,h dk (n, rk) k where (n) is the well known Mbius function and (x,y) k stands for the greatest common k th power divisor of x and y.It is immediate from lemma 2.1 of [3] that, if g(n) and h(n) are multiplicative, then S (k)(n,r) is multiplicative in both variables n and r.
For a given pair n,r of positive integers we write (resp ) to denote the largest divisor of n (resp r) that is relatively prime to r (resp n).We write n for n and for r We prove the following THEOREM.If g(n) is completely multiplicative and h(n) is multiplicative then E S(k) dk r) rk () g(r,) H (lh(p) + cklg(p) h(p) I) g,h pl r dkln k or 0 according as r, In or not, where rk(n) is the number of positive k th power divis- ors of n and the non negative integer c k ck(P) is determined so that kc k k c k+l P -and p n k r, r, We need the following lemmas.
LEMMA i.For a prime p and non negative integers a and b we have s(k)(pa, pb) b-i or u <_ b-2 according as a >_ bk, (b-l)k _< a < bk or a < (b-l)k so that the r.h.s, of (2.3) has the value as stated in (2.2).
REMARK.If g is completely multiplicative and a >_ bk >_ k we have LEMMA 2. For each integer i, 1 <_ i _< s, let t. be a non negative integer and for each ordered pair (i,j), 1 <_ i _< s, 0 <_ j <_ ti, let aij be a complex number.Then s asj (ai0 + all + + a alJl a2J2 where the summation on the left is extended over all s-tuples (jl,j 2 js with t 0-< Jii" The proof of this lemma consists of noting that each term on the left occurs exactly once in the expansion of the product on the right and vice versa.k Case (i) Suppose r, # n.In this case we have either a < k(b-1) for some prime pl or b > 1 for some prime plr.Let dkln and d H pa with 0 <_ [] Then clearly r, and hence we have either 0 _< ak < (b-l)k for some or b > i for some pl.This implies that S (k) (dk,r) 0 for each k th power divisor d k of n in virtue of lemma i and the multiplicativity of s(k)(n,r) in both variables n and r. k Case (ii) Suppose r, n.In this case a > k(b-l) for each pl and b I for each pl.Let dk In.Then d can be uniquely expressed as with k I k I and (, ) i.The multiplicativity of s(k)(n,r) in both variables implies s(k) (dk,r) s(k)(kk ) Hence we have

dk[n dk[
since, for a given k th power divisor x k of , the number of k th power divisors d k of n for which x is k(fi).Again in virtue of the multiplicativity of s(k)(n,r) in both variables we have are the prime divisors of (hence of ), Pi I' Pi I and the where PI' P2 Ps a.
summation on the right is extended over all s-tuples (i,2 s) with 0 _< i <-[]" Now lemma 2 implies that The conclusion in this case now follows in virtue of (2.5), on pushing Ih()[ into the product (since p[ ck(P) 0).

FORMULAE FOR RAMANUJAN'S SUM AND ITS GENERALIZATIONS.
Let f f(x) be a polynomial of positive degree with integer coefficients and, for positive integral r, let Nf(r) denote the number of incongruent solutions (mod r) of the congruence f(x) -= 0(rood r).Suppose such a polynomial f, a multiplicative arithme- kt tical function (n) and positive integers k and t are given Then, by taking g(n)=n and h(n) n(n) Nf (nk) (N (r) (Nf(r)) t) in (21) we have the generalized Ramanujan's This function includes as special cases (see [3]) the Rmmanujan's sum C(n,r) and some of its generalizations.In fact, writing I(n) 1 for all n and, for a given (n) exp (Hi m(n)u-I) or 0 according as n is or is not square- positive integer u u free (re(n) being the number of distinct prime factors of n) we have ck'(n r)=c(k)(n,r) X 1,I k,l (k) (Cohen [4]), Cx,t(n,r) Ct(n,r) (Cohen [53), Cx,t(n,r) C t (n,r) (M.Sugunamma [6]) i,

Bu
and C u (n r) C (n r) (C S Venkataraman and R. Sivaramakrishnan [7]) xl Specializing the functions g(n) and h(n) suitably in our theorem we obtain, for the functions described above, the following formulae: plr(l(P)Nf (Pk) + cklpkt-(P)Nf (Pk) l) k or 0 according as r, n or not Z IC u (d,r) r,() N (I + c I P-exp (Ni u-1) I) dln or 0 according as r, ln 7. ICt(d,r) r t, () n (i + cl(pt-l)) or 0 according as r,[n or not (3.6)dln plr 7.

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

PROOF
decompositions of n and r respectively.
k) (dk,r) r, k Tk() H (I + ck(pk-l)) or 0 according d In k as r, In or not (

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation