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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. [7] 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 [4] 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 [7], 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. [2], [3], [8], [7].

[section]3. Applications

1. First we prove inequality (1).

Let f (n) = S(n). Then inequalities (i), (ii) are well-known (see e.g. [1], [6], [4]). 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. [1], [6], [4]) 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 [5] 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

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

[2] 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.

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

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

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

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

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

[8] 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
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Date:Jun 1, 2012
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