# On two inequalities for the composition of arithmetic functions.

[section]1. Introduction

Let S(n) be the Smarandache (or Kempner-Smarandache) function, i.e., the function that associates to each positive integer n the smallest positive integer k such that n|k!. Let [sigma](n) denote the sum of distinct positive divisors of n, while [sigma]*(n) the sum of distinct unitary divisors of n (introduced for the first time by E. Cohen, see e.g.  for references and many informations on this and related functions). Put w(n) = number of distinct prime divisors of n, where n > 1. In paper  we have proved the inequality

S([sigma](n)) [less than or equal to] 2n - w(n), (1)

for any n > 1, with equality if and only if w(n) = 1 and 2n-1 is a Mersenne prime.

In what follows we shall prove the similar inequality

S([sigma]*(n)) [less than or equal to] n + w(n), (2)

for n > 1. Remark that n + w(n) [less than or equal to] 2n-w(n), as 2w(n) [less than or equal to] n follows easily for any n > 1. On the other hand 2n-w(n) [less than or equal to] 2n-1, so both inequalities (1) and (2) are improvements of

S(g(n)) [less than or equal to] 2n-1, (3)

where g(n) = [sigma](n) or g(n) = [sigma]*(n).

We will consider more general inequalities, for the composite functions f (g(n)), where f, g are arithmetical functions satisfying certain conditions.

[section]2. Main results

Lemma 2.1. For any real numbers a [greater than or equal to] 0 and p [greater than or equal to] 2 one has the inequality

[p.sup.a+1] - 1/p-1 [less than or equal to] [2p.sup.a] - 1, (4)

with equality only for a = 0 or p = 2.

Proof. It is easy to see that (4) is equivalent to

([p.sup.a]-1)(p-2) [greater than or equal to] 0,

which is true by p [greater than or equal to] 2 and a [greater than or equal to] 0, as [p.sup.a] [greater than or equal to] [2.sup.a] [greater than or equal to] 1 and p-2 [greater than or equal to] 0.

Lemma 2.2. For any real numbers [y.sub.i] [greater than or equal to] 2(1 [less than or equal to] i [less than or equal to] r) one has

[y.sub.1] + ... + [y.sub.r] [less than or equal to] [y.sub.1] ... [y.sub.r] (5)

with equality only for r = 1.

Proof. For r = 2 the inequality follows by ([y.sub.1]-1)([y.sub.2]-1) [greater than or equal to] 1, which is true, as [y.sub.1]-1 [greater than or equal to] 1, [y.sub.2]-1 [greater than or equal to] 1. Now, relation (5) follows by mathematical induction, the induction step [y.sub.1] ... [y.sub.r] + [y.sub.r+1] [less than or equal to] ([y.sub.1] ... [y.sub.r])[y.sub.r+1] being an application of the above proved inequality for the numbers [y'.sub.1] = [y.sub.1]... [y.sub.r], [y'.sub.2] = [y.sub.r+1].

Now we can state the main results of this paper.

Theorem 2.1. Let f, g : N [right arrow] R be two arithmetic functions satisfying the following conditions:

(i) f (xy) [less than or equal to] f (x) + f (y) for any x,y [member of] N.

(ii) f (x) [less than or equal to] x for any x [member of] N.

(iii) g([p.sup.[alpha]]) [less than or equal to] [2p.sup.[alpha]] -1, for any prime powers [p.sup.[alpha]] (p prime, [alpha] [greater than or equal to] 1).

(iv) g is multiplicative function.

Then one has the inequality

f (g(n)) [less than or equal to] 2n-w(n),n> 1. (6)

Theorem 2.2. Assume that the arithmetical functions f and g of Theorem 2.1 satisfy conditions (i), (ii), (iv) and

(iii)' g([p.sup.[alpha]]) [less than or equal to] [p.sup.[alpha]] + 1 for any prime powers [p.sup.[alpha]].

Then one has the inequality

f (g(n)) [less than or equal to] n + w(n), n> 1. (7)

Proof of Theorem 2.1. As f([x.sub.1]) [less than or equal to] f([x.sub.1]) and

f([x.sub.1][x.sub.2]) [less than or equal to] f([x.sub.1]) + f([x.sub.2]),

it follows by mathematical induction, that for any integers r [greater than or equal to] 1 and [x.sub.1], ... , [x.sub.r] [greater than or equal to] 1 one has

f ([x.sub.i] ... [x.sub.r]) [less than or equal to] f([x.sub.1]) + ... + f([x.sub.r]). (8)

Let now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 be the prime factorization of n, where [p.sub.i] are distinct primes and [[alpha].sub.i] [greater than or equal to] 1 (i = 1, ... , r). Since g is multiplicative, by inequality (8) one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By using conditions (ii) and (iii), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by Lemma 2.2 we get inequality (6), as r = w(n).

Proof of Theorem 2.2. Use the same argument as in the proof of Theorem 2.1, by remarking that by (iii)'

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 2.1. By introducing the arithmetical function [B.sup.1] (n) (see , Ch.IV.28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(i.e., the sum of greatest prime power divisors of n), the following stronger inequalities can be stated:

f (g(n)) [less than or equal to] [2B.sup.1] (n)-w(n), (6')

(in place of (6)); as well as:

f (g(n)) [less than or equal to] [B.sup.1] (n) + w(n), (7')

(in place of (7)).

For the average order of [B.sup.1](n), as well as connected functions, see e.g. , , , .

[section]3. Applications

1. First we prove inequality (1).

Let f (n) = S(n). Then inequalities (i), (ii) are well-known (see e.g. , , ). Put g(n) = f[sigma](n). As [sigma]([p.sup.[alpha]]) = [p.sup.[alpha]+1]-1/p-1, inequality (iii) follows by Lemma 2.1. Theorem 2.1 may be applied.

2. Inequality (2) holds true.

Let f (n) = S(n), g(n) = [sigma]*(n). As [sigma]*(n) is a multiplicative function, with [sigma]*([p.sup.[alpha]]) = [p.sup.[alpha]]+ 1, inequality (iii)' holds true. Thus (2) follows by Theorem 2.2.

3. Let g(n) = [psi](n) be the Dedekind arithmetical function, i.e., the multiplicative function whose value of the prime power [p.sup.[alpha]] is

[psi] [p.sup.[alpha]])= [p.sup.[alpha]-1](p +1).

Then [psi]([p.sup.[alpha]]) [less than or equal to] 2[p.sup.[alpha]]-1 since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is true, with strict inequality.

Thus Theorem 2.1 may be applied for any function f satisfying (i) and (ii).

4. There are many functions satisfying inequalities (i) and (ii) of Theorems 2.1 and 2.2. Let f (n) = log [sigma](n).

As [sigma](mn) [less than or equal to] [sigma](m)[sigma](n) for any m, n [greater than or equal to] 1, relation (i) follows. The inequality f(n) [less than or equal to] n follows by [sigma](n) [less than or equal to] [e.sup.n], which is a consequence of e.g. [sigma](n) [less than or equal to] [n.sup.2] < [e.sup.n] (the last inequality may be proved e.g. by induction).

Remark 3.1. More generally, assume that F(n) is a submultiplicative function, i.e., satisfying

F(mn) [less than or equal to] F(m)F(n) for m, n [greater than or equal to] 1. (i')

Assume also that

F(n) [less than or equal to] [e.sup.n]. (ii')

Then f (n) = log F(n) satisfies relations (i) and (ii).

5. Another nontrivial function, which satisfies conditions (i) and (ii) is the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly, f(n) [less than or equal to] n, with equality only if n = 1 or n = prime. For y = 1 we get f(x) [less than or equal to] f(x) + 1 = f(x) + f(1), when x, y [greater than or equal to] 2 one has

f (xy) = 1 [less than or equal to] f (x) + f (y).

6. Another example is

f(n) = [OMEGA](n)= [[alpha].sub.1] + ... + [[alpha].sub.r], (10)

for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.e., the total number of prime factors of n. Then f (mn) = f (m) + f(n), as [OMEGA](mn) = [OMEGA](m) + [OMEGA](n) for all m, n [greater than or equal to] 1. The inequality [OMEGA](n) < n follows [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

7. Define the additive analogue of the sum of divisors function [sigma], as follows: If n = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the prime factorization of n, put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

As [sigma](n) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], clearly by Lemma 2.2 one has

[SIGMA](n) [less than or equal to] [sigma](n). (12)

Let f(n) be any arithmetic function satisfying condition (ii), i.e., f(n) [less than or equal to] n for any n [greater than or equal to] 1. Then one has the inequality:

f ([SIGMA](n)) [less than or equal to] 2[B.sup.1] (n)-w(n) [less than or equal to] 2n-w(n) [less than or equal to] 2n-1 (13)

for any n > 1.

Indeed, by Lemma 2.1, and Remark 2.1, the first inequality of (13) follows. Since [B.sup.1](n) [less than or equal to] n (by Lemma 2.2), the other inequalities of (13) will follow. An example:

S([SIGMA](n)) [less than or equal to] 2n-1, (14)

which is the first and last term inequality in (13).

It is interesting to study the cases of equality in (14). As S(m) = m if and only if m = 1, 4 or p (prime) (see e.g. , , ) and in Lemma 2.2 there is equality if w(n) = 1, while in Lemma 2.1, as p = 2, we get that n must have the form n = [2.sup.[alpha]]. Then [SIGMA](n) = [2.sup.[alpha]+1]-1 and [2.sup.[alpha]+1]-1 [not equal to] 1, [2.sup.[alpha]+1]-1 [not equal to] 4, [2.sup.[alpha]+1]-1 = prime, we get the following theorem:

There is equality in (14) iff n = [2.sup.[alpha]], where [2.sup.[alpha]+1] -1 is a prime.

In paper  we called a number n almost f-perfect, if f(n) = 2n-1 holds true. Thus, we have proved that n is almost S o [SIGMA]-perfect number, iff n = [2.sup.[alpha]], with [2.sup.a+1] -1 a prime (where "o" denotes composition of functions).

References

 Ch. Ashbacher, An introduction to the Smarandache function, Erhus Univ. Press, Vail, 1995.

 P. Erdos and A. Ivic, Estimates for sums involving the largest prime factor of an integer and certain related additive functions, Studia Sci. Math. Hung., 15(1980), 183-199.

 J. M. de Koninck and A. Ivic, Topics in arithmetical functions, Notas de Matematica, North Holland, 1980.

 F. Luca and J. Sandor, On the composition of a certain arithmetic function, Functiones et Approximatio, 41(2009), 185-209.

 J. Sandor, On the composition of some arithmetic functions, Studia Univ. Babes-Bolyai, Math., 34(1989), 7-14.

 J. Sandor, Geometric theorems, diophantine equations and arithmetic functions, New Mexico, 2002.

 J. Saandor, D. S. Mitrinoviac and B. Crstici, Handbook of number theory I, Springer Verlag, 2006.

 T. Z. Xuan, On some sums of large additive number theoretic functions (Chinese), Journal of Beijing normal university, 1984, No. 2, 11-18.

Jozsef Sandor

Department of Mathematics, Babes-Bolyai University, Str. Kogalniceanu nr. 1, 400084 Cluj-Napoca, Romania

E-mail: jsandor@math.ubbcluj.ro