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Sums over Numbers with Restricted Prime Factors

Khadija Mbarki, Walid Wannes*

Faculté des sciences de Sfax, BP 1171, Sfax 3000, Tunisie

Abstract: Let be a positive integer and be the sum of the digits in basis q of the positive integer n. We prove that the quotient has a normal order one, where and are respectively, the number of distinct prime factors and the number of prime factors p of a positive integer n counted with multiplicity such that mod Moreover, we discuss sums of the form where f is a multiplicative function.

Keywords: sum-of-digits function, uniform distribution modulo1, multiplicative function

Received May 12, 2017; Revised June 30, 2107; Accepted July 12, 2017

Citation: Khadija Mbarki, Walid Wannes, Sums over Numbers with Restricted Prime Factors. Turkish Journal of Analysis and Number Theory, 5 (5). 153-158. January 2017. doi:10.12691/tjant-5-5-2

1. Introduction

Let be a fixed integer. Every positive integer n has a unique q-adic expansion of the form

with For we admit to set The sum of digits function in basis q is defined by

The function S is one of the most natural examples of q-additive functions (i.e. with ) and is the set of non negative integers. Such functions were introduced by Gelfond [7] and further studied by Coquet [1], Kátai [9] and others.

For every and such that we define as the number of distinct prime factors p of n such that mod b and the number of distinct prime factors of n counted with multiplicity such that mod b. Therefore,

Both functions have been studied in [12,13], it was proved that their normal order is A crucial part in the proof is the following estimation

[picture] | (1.1)

where

The aim of this paper is to provide asymptotic formulas for sums involving and We consider firstly the sum

Then, by using elementary methods, we discuss sums of the form

where f is one of the classical arithmetic functions , and where is the Möbius function, while and are respectively, the Euler function and the sum of divisors function. This extends the results known through the work of De Koninck [4], De Koninck and Sitaramachandrarao [5] to primes verifying a digital constraint. Finally, we give some related results about the distribution of and uniform distribution modulo 1 of sequences involving and

Throughout this paper, always denotes a prime number. For any real x, we set The notation refers to the greatest common divisor of a and b. is the number of distinct prime factors of n while is the number of prime factors counted with multiplicity of the integer n. We recall that the notation is equivalent to the statement that for positive functions U and V and the implied constants in the symbols “O”, “” are absolute. We also use the symbol “o” with its usual meaning, the statement is equivalent to

2. The Sum of Quotients

Since and have both the normal order (see [8]), it is obvious that has normal order one which is shown by De Koninck in [4]. A crucial part in his proof is an elementary estimation of In this part, we consider an analogue problem, since and have both the normal order (see [12]). So, we put

Then, the average value of is given in the following theorem.

Theorem 2.1.

[picture] | (2.1)

Proof. Our first task is to estimate Indeed, we have

Proposition 2.2 of [12] implies that

where C is an absolute constant. So,

Or,

Then,

So, we obtain

[picture] | (2.2)

Now, we use (2.2) to estimate the left hand side of (2.1) as follows

[picture] | (2.3)

Since then by partial summation, we can write

[picture] | (2.4)

where the last equality is derived from (2.2). Assembling (2.3) and (2.4), we obtain

which completes the proof of the theorem.

3. On the Sum

In this section, we discuss sums of the form where is either or We note that the case where f is a multiplicative function, was discussed in [[6], Chapter 9].

Theorem 3.1. Let q, b be integers satisfying Then for any integer there exist constants such that

Proof.

[picture] | (3.1)

By pursuing the same procedure, we get

Now, we write

[picture] | (3.2)

say.

For the estimation of we use Dirichlet’s hyperbola method and we obtain

[picture] | (3.3)

Now, it is well-known (See [15]) that there exists a constant A > 0 such that

[picture] | (3.4)

Hence,

[picture] | (3.5)

where the last bound is derived from (1.1). For the estimations of and we write

[picture] | (3.6)

for any c > 0, by using the prime number theorem and [[11], Théorème 3] in the case Here Also

[picture] | (3.7)

However, by partial summation, we derive that

[picture] | (3.8)

Now,

[picture] | (3.9)

From (3.7), (3.8) and (3.9), we deduce

Assembling (3.5), (3.6) and the last equality, one can show that

[picture] | (3.10)

with

For by using (3.4), we get

[picture] | (3.11)

Finally, formulas (3.2), (3.10) and (3.11) give the desired estimation.

In order to provide a corresponding result for φ and σ, we need the lemma below, which can be proved by the same method as in the proof provided in [13].

Lemma 3.2. Let and Then, for any integer we have

Theorem 3.3. Let q, b be integers verifying Then for any integer there exist constants K₁, K₂, Fi, Gi, such that

Proof. For the Euler function, if we proceed as in (3.1), we obtain

[picture] | (3.12)

Since is a multiplicative function verifying one can show that

[picture] | (3.13)

By using the same approach as in Theorem 3.1, Lemma 3.2 and the following well known formulas

combined with (3.12) and (3.13), we get the result.

4. Other Results

4.1. The Distribution of

Rényi proved that for any positive integer k, the set of numbers n such that has density where are the power series coefficients of the meromorphic function

Let denote the number of for which In [[14], Theorem 2.16], Montgomery and Vaughan showed a quantitative form of Rényi’s theorem which states that for any nonnegative integer and any , one has

[picture] | (4.1)

where

and is the set of powerful numbers i.e., those m such that In this section, we will study the distribution of the function An analogous formula to (4.1) holds for the number of for which Denote this number. By following the same steps as in the proof of [[14], Theorem 2.16], we get the following result.

Proposition 4.1. For any nonnegative integer k and any

where

and is the set defined above.

4.2. Uniform Distribution Modulo 1

We denote by the set of multiplicative arithmetical functions verifying

H. Daboussi proved that for every irrational uniformly for in we have

The proof is given in his paper [2] written jointly with H. Delange.

An immediate consequence of Daboussi’s result is the following: If is an irrational number and g is a real valued additive arithmetical function, then the sequence is uniformly distributed modulo 1 and this follows from Weyl’s criterion (see [[10], Theorem 5.6]). Since and are both additive real valued arithmetical functions then, for any irrational number the following sequences and are uniformly distributed modulo 1.

References

[1]   | J. Coquet. Sur les fonctions Q-multiplicatives et Q-additives, Thèse 3ème cycle, Orsay, (1975).
[2]   | H. Daboussi, H. Delange. On multiplicative arithmetical functions whose module does not exceed one, J. London Math. Soc. 26, p.245-264, (1982).
[3]   | M. Drmota, C. Mauduit, J. Rivat. Primes with an average sum of digits, Compositio Math. 145, p.271-292, (2009).
[4]   | J. M. De Koninck. Sums of quotients of additive functions, Proc. Amer. Math. Soc, (44), p. 35-38, (1974).
[5]   | J. M. De Koninck and R. Sitaramachandrarao. Sums involving the largest prime divisor of an integer II, Indian J. pure appl. Math., (19), p. 990-1004, (1988).
[6]   | J. M. De Koninck and A. Ivić, Topics in Arithmetical functions Notas de Matematica 72, North Holland, Amsterdam, (1980).
[7]   | A. O. Gelfond. Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. (13), p. 259-265, (1968).
[8]   | G.H. Hardy and E. M. Wright. An introduction to the theory of numbers, Oxford University Press, (1979).
[9]   | I. Kàtai. Distribution of q-additive functions, in “Probability Theory and Applications” Kluwer Academic, Dordrecht, p. 309-318, (1992).
[10]   | L. Kuipers, H. Niederreiter. Uniform distribution of sequences John Willey, New York, (1974).
[11]   | C. Mauduit, J.Rivat. Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. 71, p.159-1646, (2010).
[12]   | M. Mkaouar, W. Wannes. On the number of restricted prime factors of an integer, Acta Mathematica Hungarica, (143), p. 88-95, (2014).
[13]   | M. Mkaouar, W. Wannes. On the normal number of prime factors of φ(n) subject to certain congruence conditions, J. of Number Theory (160), p. 629-645, (2016).
[14]   | H.L. Montgomery, R.C. Vaughan. Multiplicative Number Theory I. Classical Theory Cambridge Studies in advanced Mathematics 97.
[15]   | A. Walfisz. Weylsche Exponentialsummen in der neuren Zahlentheorie, Berlin, (1963).