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volume 6, issue 2, article 31, 2005.

Received 05 October, 2004;

accepted 16 February, 2005.

Communicated by:L. Tóth

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Journal of Inequalities in Pure and Applied Mathematics

A NOTE ON SUMS OF POWERS WHICH HAVE A FIXED NUMBER OF PRIME FACTORS

RAFAEL JAKIMCZUK

Department of Mathematics Universidad Nacional de Luján Argentina

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 182-04

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A Note On Sums Of Powers Which Have A Fixed Number Of

Prime Factors Rafael Jakimczuk

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J. Ineq. Pure and Appl. Math. 6(2) Art. 31, 2005

http://jipam.vu.edu.au

Abstract

Let us denote byc_n,k the sequence of numbers which have in its factorization kprime factors (k≥1), we obtain in short proofs asymptotic formulas forc_n,k, Pn

i=1c^α_i,kandP

ci,k≤xc^α_i,k. We generalize the work by T. Sálat y S. Znam when k= 1(see reference [2]).

2000 Mathematics Subject Classification:11N25, 11N37.

Key words: Sums of powers, Numbers withkprime factors.

Letπ_k(x)be the number of these numbers not exceedingx, it was proved by Landau [1] that

(1) lim

x→∞

π_k(x)

x(log logx)^k−1 (k−1)! logx

= 1

Note that if k = 1thenπ₁(x) = π(x),c_n,1 =p_n, and equation (1) is the prime number theorem.

Theorem 1. The following asymptotic formula holds:

(2) c_n,k ∼ (k−1)!nlogn

(log logn)^k−1 .

(3)

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Proof. Ifk = 1the formula is true, since in this case (2) is the prime number theorempn ∼ nlogn. Suppose k ≥ 2. If we putx = cn,k and substitute into (1) we find that

(3) lim

n→∞

(k−1)!nlogc_n,k

c_n,k (log logc_n,k)^k−1 = 1.

Writing

(4) c_n,k = (k−1)!nlogn

(log logn)^k−1 f(n) and substituting (4) into (3) we obtain

(5) lim

n→∞

logc_n,k(log logn)^k−1

logn f(n) (log logc_n,k)^k−1 = 1.

From equation (1) we find that

(6) lim

x→∞

π_k(x) π(x) =∞.

Assume that the inequalities c_n,k ≥ p_n have infinitely many solutions, then we have π(c_n,k) ≥ π(p_n) = n = π_k(c_n,k), which contradicts (6). Hence for all sufficiently large n we have c_n,k < p_n. On the other hand, clearly n ≤ c_n,k. Thereforen≤c_n,k≤p_n, that islogn ≤logc_n,k ≤logp_n, and we find that

(7) 1≤ logc_n,k

logn ≤ logp_n logn .

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From (7) and the prime number theoremp_n∼nlogn, we obtain

(8) lim

n→∞

logc_n,k logn = 1.

From (5) and (8) we find that

(9) lim

n→∞f(n) = 1.

To finish, (9) and (4) give (2). The theorem is thus proved.

The following proposition is well known, we use it as a lemma Lemma 2. Let P∞

i=1ai and P∞

i=1bi be two series of positive terms such that

n→∞lim

an

bn = 1. Then ifP∞

i=1b_i is divergent, the following limit holds

n→∞lim Pn

i=1a_i Pn

i=1bi

= 1.

Theorem 3. Letk≥1and letαbe a positive number. The following asymptotic formula holds

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n

X

i=1

c^α_i,k ∼ ((k−1)!)^αn^α+1log^αn (α+ 1) (log logn)^α(k−1). Proof. Let us consider the following two series:

∞

X

i=1

c^α_i,k and 1 + 2 +

∞

X

i=3

(k−1)! ilogi (log logi)^k−1

!α

.

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Since the function_(k−1)!

tlogt (log logt)^k−1

α

is increasing from a certain value oft, we find that

(11) 1 + 2 +

n

X

i=3

(k−1)!ilogi (log logi)^k−1

!α

= Z n

3

(k−1)!tlogt (log logt)^k−1

!α

dt+O

nlogn (log logn)^k−1

!α! .

On the other hand, from the L’Hospital rule (12)

Z n 3

(k−1)!tlogt (log logt)^k−1

!α

dt∼ ((k−1)!)^αn^α+1log^αn (α + 1) (log logn)^α^(k−1). Equation (10) is an immediate consequence of (11), (12) and the lemma.

The theorem is thus proved.

Theorem 4. Letk≥1and letαbe a positive number. The following asymptotic formula holds

(13) X

ci,k≤x

c^α_i,k ∼ x^α+1(log logx)^k−1 (α+ 1) (k−1)! logx. Proof. Equation (3) can be written in the form

(14) lim

n→∞

n

cn,k(^{log log}^cn,k)^k−1

(k−1)! logcn,k

= 1.

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From (8) we obtain

(15) lim

n→∞

log logc_n,k log logn = 1.

Substituting (14), (8) and (15) into (10) we find that

(16) X

c_i,k≤c_n,k

c^α_i,k ∼ c^α+1_n,k (log logc_n,k)^k−1 (α+ 1) (k−1)! logcn,k

.

Equation (2) givesc_n,k ∼c_n+1,k, therefore X

c_i,k≤c_n,k

c^α_i,k ∼ c^α+1_n+1,k(log logc_n+1,k)^k−1 (α+ 1) (k−1)! logcn+1,k

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∼ c^α+1_n,k (log logc_n,k)^k−1 (α+ 1) (k−1)! logcn,k

. Since the function

x^α+1(log logx)^k−1 (α+ 1) (k−1)! logx

is increasing from a certain value ofx, we have for allnsufficiently large (18)

P

ci,k≤cn,k

c^α_i,k

c^α+1_n,k (log logcn,k)^k−1 (α+1) (k−1)! logcn,k

≤

P

ci,k≤x

c^α_i,k

x^α+1(log logx)^k−1 (α+1) (k−1)! logx

6

P

ci,k≤cn,k

c^α_i,k

c^α+1_n+1,k(log logcn+1,k)^k−1 (α+1) (k−1)! logcn+1,k

,

wherec_n,k ≤x < c_n+1,k.

To finish, (17) and (18) give (13). The theorem is proved.

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Note. The casek = 1was studied in the reference [2]. In this case (9) and (13) become

n

X

i=1

p^α_i ∼ n^α+1log^αn

(α+ 1) , X

pi≤x

p^α_i ∼ x^α+1 (α+ 1) logx.

Using equation (2) and the lemma, we can prove (as above) other theorems, for example the following:

Theorem 5. The following asymptotic formulas holds

∞

X

n=1

1

c_n,k ∼ (log logn)^k

k! and X

cn,k≤x

1

c_n,k ∼ (log logx)^k k! . Whenk = 1, this theorem is well known.

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References

[1] G.H. HARDYANDE.M. WRIGHT, An Introduction to Number Theory, 4th Ed.1960. Chapter XXII.

[2] T. SÁLAT AND S. ZNAM, On the sums of prime powers, Acta Fac. Rer.

Nat. Univ. Com. Math., 21 (1968), 21–25.