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.
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
![]() | (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
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.
![]() | (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
![]() | (2.2) |
Now, we use (2.2) to estimate the left hand side of (2.1) as follows
![]() | (2.3) |
Since then by partial summation, we can write
![]() | (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.
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.
![]() | (3.1) |
By pursuing the same procedure, we get
![]() |
Now, we write
![]() | (3.2) |
say.
For the estimation of we use Dirichlet’s hyperbola method and we obtain
![]() | (3.3) |
Now, it is well-known (See 15) that there exists a constant A > 0 such that
![]() | (3.4) |
Hence,
![]() | (3.5) |
where the last bound is derived from (1.1). For the estimations of and
we write
![]() | (3.6) |
for any c > 0, by using the prime number theorem and [ 11, Théorème 3] in the case Here
Also
![]() | (3.7) |
However, by partial summation, we derive that
![]() | (3.8) |
Now,
![]() | (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
![]() | (3.10) |
with
![]() |
For by using (3.4), we get
![]() | (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
![]() | (3.12) |
Since is a multiplicative function verifying
one can show that
![]() | (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.
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
![]() | (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.
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.
[1] | J. Coquet. Sur les fonctions Q-multiplicatives et Q-additives, Thèse 3ème cycle, Orsay, (1975). | ||
In article | |||
[2] | H. Daboussi, H. Delange. On multiplicative arithmetical functions whose module does not exceed one, J. London Math. Soc. 26, p.245-264, (1982). | ||
In article | View Article | ||
[3] | M. Drmota, C. Mauduit, J. Rivat. Primes with an average sum of digits, Compositio Math. 145, p.271-292, (2009). | ||
In article | View Article | ||
[4] | J. M. De Koninck. Sums of quotients of additive functions, Proc. Amer. Math. Soc, (44), p. 35-38, (1974). | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[6] | J. M. De Koninck and A. Ivić, Topics in Arithmetical functions Notas de Matematica 72, North Holland, Amsterdam, (1980). | ||
In article | |||
[7] | A. O. Gelfond. Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. (13), p. 259-265, (1968). | ||
In article | View Article | ||
[8] | G.H. Hardy and E. M. Wright. An introduction to the theory of numbers, Oxford University Press, (1979). | ||
In article | View Article | ||
[9] | I. Kàtai. Distribution of q-additive functions, in “Probability Theory and Applications” Kluwer Academic, Dordrecht, p. 309-318, (1992). | ||
In article | View Article | ||
[10] | L. Kuipers, H. Niederreiter. Uniform distribution of sequences John Willey, New York, (1974). | ||
In article | |||
[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). | ||
In article | View Article | ||
[12] | M. Mkaouar, W. Wannes. On the number of restricted prime factors of an integer, Acta Mathematica Hungarica, (143), p. 88-95, (2014). | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[14] | H.L. Montgomery, R.C. Vaughan. Multiplicative Number Theory I. Classical Theory Cambridge Studies in advanced Mathematics 97. | ||
In article | View Article | ||
[15] | A. Walfisz. Weylsche Exponentialsummen in der neuren Zahlentheorie, Berlin, (1963). | ||
In article | View Article | ||
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[1] | J. Coquet. Sur les fonctions Q-multiplicatives et Q-additives, Thèse 3ème cycle, Orsay, (1975). | ||
In article | |||
[2] | H. Daboussi, H. Delange. On multiplicative arithmetical functions whose module does not exceed one, J. London Math. Soc. 26, p.245-264, (1982). | ||
In article | View Article | ||
[3] | M. Drmota, C. Mauduit, J. Rivat. Primes with an average sum of digits, Compositio Math. 145, p.271-292, (2009). | ||
In article | View Article | ||
[4] | J. M. De Koninck. Sums of quotients of additive functions, Proc. Amer. Math. Soc, (44), p. 35-38, (1974). | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[6] | J. M. De Koninck and A. Ivić, Topics in Arithmetical functions Notas de Matematica 72, North Holland, Amsterdam, (1980). | ||
In article | |||
[7] | A. O. Gelfond. Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith. (13), p. 259-265, (1968). | ||
In article | View Article | ||
[8] | G.H. Hardy and E. M. Wright. An introduction to the theory of numbers, Oxford University Press, (1979). | ||
In article | View Article | ||
[9] | I. Kàtai. Distribution of q-additive functions, in “Probability Theory and Applications” Kluwer Academic, Dordrecht, p. 309-318, (1992). | ||
In article | View Article | ||
[10] | L. Kuipers, H. Niederreiter. Uniform distribution of sequences John Willey, New York, (1974). | ||
In article | |||
[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). | ||
In article | View Article | ||
[12] | M. Mkaouar, W. Wannes. On the number of restricted prime factors of an integer, Acta Mathematica Hungarica, (143), p. 88-95, (2014). | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[14] | H.L. Montgomery, R.C. Vaughan. Multiplicative Number Theory I. Classical Theory Cambridge Studies in advanced Mathematics 97. | ||
In article | View Article | ||
[15] | A. Walfisz. Weylsche Exponentialsummen in der neuren Zahlentheorie, Berlin, (1963). | ||
In article | View Article | ||