Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 5, Article 70, 2002
ON SHORT SUMS OF CERTAIN MULTIPLICATIVE FUNCTIONS
OLIVIER BORDELLÈS
22 RUE J EAN BARTHÉLEMY
43000 LE PUY-EN-VELAY FRANCE
borde43@wanadoo.fr
Received 30 August, 2002; accepted 16 October, 2002
Communicated by L. Toth
A BSTRACT. We use the recent theory of integer points close to a smooth curve developed by
Huxley-Sargos and Filaseta-Trifonov to get an asymptotic formula for short sums of a class of
multiplicative functions.
Key words and phrases: Multiplicative Functions, Short Sums, Integer points close to a curve.
2000 Mathematics Subject Classification. 11N37, 11P21.
1. I NTRODUCTION
AND
N OTATION
Let k > 2 be an integer. A positive integer n is said to be k−free (resp. k−full) if, for any
prime p | n, the p−adic valuation vp (n) of n satisfies vp (n) < k (resp. vp (n) > k), and we
use the terms squarefree or squarefull when k = 2. We denote by µk the multiplicative function
defined by

 1, if n is k − free
µk (n) :=

0, otherwise.
Obtaining gap results for k−free (or k−full) numbers is a very famous problem in analytic
number theory (see [1] and the references). The best estimation in this direction has been
obtained by Filaseta and Trifonov ([1]) who showed that, for x sufficiently large, any interval
of the type x; x + cx1/(2k+1) log x (c := c (k) > 0) contains a k−free number.
A dual problem is to get an asymptotic formula for µk . This requires estimations for short
sums of multiplicative functions, but such results are still relatively rare in the literature (see
[2, 6]). In this paper, we are motivated by finding asymptotic results for short sums of the
following class of arithmetical functions: define M to be the set of multiplicative functions f
verifying 0 6 f (n) 6 1 for any positive integer n and f (p) = 1 for any prime number p. If
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
092-02
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O LIVIER B ORDELLÈS
f ∈ M, we set
Y
P (f ) :=
p
1
1−
p
∞
X
f pl
1+
pl
l=1
!
.
We prove:
Theorem 1.1. Let ε > 0 and x, y > 2 be real numbers such that x6ε 6 y 6 x. If f ∈ M, we
have as x → +∞ :
X
f (n) = yP (f ) + O x1/6+ε + y 1/2 .
x<n6x+y
Moreover, if y 6 x1/3 , then:
X
f (n) = yP (f ) + O x1/7+ε + x1/21+ε y 2/3 .
x<n6x+y
For example, using this, we get:
X
µ2 (n) =
x<n6x+y
6y
+ O x1/6+ε + y 1/2
2
π
with x6ε 6 y 6 x. The proof of this theorem uses a convolution argument, and the new theory of
integer points close to a curve (see [1, 3, 4]) arises as the crucial point to estimate the difference
of integer parts.
In what follows, a, b, d, k, l, m, n, q, B, N will always denote positive integers, p will be a
prime number and [x] is the integral part of x. For any X, Y > 0, the notation X Y means
there exists C > 0 such that X 6 CY. the notation X Y means X Y and Y X
simultaneously.
If f, g are two arithmetical functions, the Dirichlet convolution product f ∗ g is defined by
X
(f ∗ g) (n) :=
f (d) g (n/d) .
d|n
P
µ (n) is the Möbius function, τ (n) := d|n 1 the classical divisor function, and, more generally,
τ(k) (n) is the arithmetical function defined by
X
τ(k) (n) :=
1.
dk |n
If ϕ is any smooth function and if δ > 0 is any real number, we set
R (ϕ, N, δ) := |{n ∈ ]N ; 2N ] ∩ Z, ∃ m ∈ Z, |ϕ (n) − m| 6 δ}| .
Thus, if δ is sufficiently small, R (ϕ, N, δ) counts the number of integer points close to the
curve y = ϕ (x) when N < x 6 2N, and then
X
R (ϕ, N, δ) =
([ϕ (n) + δ] − [ϕ (n) − δ]) .
N <n62N
We will therefore use the following principle: let M be a large real number, ϕ (n) and δ (n)
two functions such that δ := maxn6M δ (n) satisfies 0 < δ 6 1/4. Then we have
X
(1.1)
([ϕ (n) + δ (n)] − [ϕ (n)]) max R (ϕ, N, δ) log M.
n6M
J. Inequal. Pure and Appl. Math., 3(5) Art. 70, 2002
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S HORT S UMS
3
2. AUXILIARY R ESULTS
We collect here the lemmas we will use to show the theorem. The first was proved by Huxley
and Sargos in [3, 4] using divided differences and the reduction principle, and the second is due
to a remarkable work by Filaseta and Trifonov ([1, Theorem 7]) who used divided differences
and a very useful polynomial identity:
Lemma 2.1. Let δ > 0 be a real number and ϕ ∈ C k (]N ; 2N ] 7→ R) such that there exists a
real number λk > 0 such that
(k)
ϕ (x) λk
for any real number x ∈ ]N ; 2N ] . Then:
(i) If k > 2,
2
1/k
2
+ 1.
R (ϕ, N, δ) N λkk(k+1) + N δ k(k−1) + δλ−1
k
(k−1)
(ii) Moreover, if k > 5 and ϕ
(x) λk−1 = N λk for any real number x ∈ ]N ; 2N ] ,
then:
2
1
k−1
2
R (ϕ, N, δ) N λkk(k+1) + N δ (k−1)(k−2) + δλ−1
+ 1.
k−1
Lemma 2.2. Let k > 2 be an integer and x, c0 > 0, δ > 0 be real numbers (c0 sufficiently
small) satisfying N k−1 δ 6 c0 and N 6 x1/k . Then we have:
x
6k2 +k−1
1
1
R k , N, δ x 2k+1 + x 6k+3 δN 6k+3 .
n
Lemma 2.3. Let f ∈ M and z > 1 any real number. Then:
(i)
X
X 1
< 8z −1/2 .
1 < 3z 1/2 and
n
n>z
n6z
n squarefull
n squarefull
(ii)
X
|(f ∗ µ) (n)| < 4z 1/2
and
n6z
Proof.
X |(f ∗ µ) (n)|
< 8z −1/2 .
n
n>z
(i) Since every squarefull number n can be written in a unique way as n = a2 b3
with b squarefree, we have:
∞
X
X
X
X
3
1/2
−3/2
16
1<z
b
=ζ
z 1/2
2
√
1/3
n6z
b=1
−3
b6z
a6 zb
n squarefull
and the well-known inequality ζ (σ) 6 σ/ (σ − 1) gives the first part of the result. In
the same way, let Z > z be any real number. We have:
X 1
X 1
X
X
1
1 X 1
6
+
n
b3 √ −3
a2
b3 √ −3 a2
√
1/3
1/3
1/3
z<n6Z
−3
b6z
zb
z
<a6 Zb
<b6Z
a6 Zb
n squarefull
< 2z −1/2
X
b6z 1/3
6 6z −1/2 +
J. Inequal. Pure and Appl. Math., 3(5) Art. 70, 2002
1
b3/2
+
π2 X 1
6
b3
1/3
b>z
2
π −2/3
z
< 8z −1/2 .
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4
O LIVIER B ORDELLÈS
(ii) We set g := f ∗ µ. The hypothesis f (p) = 1 implies |g (p)| = 0 and, using multiplicativity, |g (n1 )| = 0 for any positive squarefree integer n1 > 1. Since any positive integer
n can be written in a unique way as n = n1 n2 with n1 squarefree, n2 squarefull and
(n1 , n2 ) = 1, we deduce that |g (n)| =
6 0 if either n = 1 or n > 1 is squarefull. The
result follows by using (i) and the fact that |g (n)| 6 1 for any positive integer n.
Lemma 2.4. Let k > 1 be an integer and ε > 0 be a fixed real number. Then, for any positive
integer d, we have:
21/ε
2
τ(k) (d) 6
dε/k .
eε log 2
1/ε
Proof. We set c (ε) := 22 (eε log 2)−2 . The bound τ (d) 6 c (ε) dε is well-known (see [5]).
Since any positive integer n can be represented in a unique way as n = qmk with q a positive
k−free integer, we have
1/ε
τ(k) (d) = τ (m) 6 c (ε) mε 6 c (ε) dε/k .
3. P ROOF OF THE T HEOREM
Let ε > 0 be a fixed real number. We take g := f ∗ µ again, and we have:
X
X X
f (n) =
g (d)
x<n6x+y
x<n6x+y d|n
X
=
g (d)
d6x+y
=
X
d6y
X
+
h i
x
x+y
−
d
d
:= Σ1 + Σ2 .
y<d6x+y
(1) Using Lemma 2.3, we have:
Σ1 = y
X g (d)
d6y
d
!
+O
X
|g (d)|
d6y
!
X |g (d)|
g (d)
=y
+O y
+ O y 1/2
d
d
d=1
d>y
!
∞
Y
X
f pl − f pl−1
=y
1+
+ O y 1/2
l
p
p
l=1
!
∞
Y
1 X f pl
1
1/2
=y
1+ 1−
−
+
O
y
p l=1 pl
p
p
= yP (f ) + O y 1/2 .
∞
X
(2) Writing again d = a2 b3 with µ2 (b) = 1, we have:
h i
X
x+y
x
|Σ2 | 6
|g (d)|
−
d
d
y<d6x+y
d squarefull
J. Inequal. Pure and Appl. Math., 3(5) Art. 70, 2002
http://jipam.vu.edu.au/
S HORT S UMS
6
X
b6(x+y)1/3
=
√y
<a6
b3
X
b6(x+y)
X
q
x+y
b3
5
−3 (x + y) b−3
xb
−
2
a
a2
X
X
1/3 xb−3 <d6(x+y)b−3
√y
<a6
b3
6
X
b6(x+y)
X
1/3
1
a2 |d
q
x+y
b3
τ(2) (d)
xb−3 <d6(x+y)b−3
h i
x+y
x
6 c (ε) (2x)
− 3
3
b
b
b6(x+y)1/3
x
y xε
max
R 3 , B, 3 ,
b
B
16B6(x+y)1/3
ε/2
X
where we used (1.1) , Lemma 2.4 (with k = 2) and the inequality log x 6 2 (eε)−1 xε/2 .
Now Lemma 2.1(i) with k = 3 and λ3 = xB −6 gives
|Σ2 | xε x1/6 + y 1/3 x1/6+ε + y 1/2
since y > x6ε .
(3) We suppose now that y 6 x1/3 . One can improve the former estimation by using Lemma
2.2 instead of Lemma 2.1. The hypothesis N k−1 δ 6 c0 compels us to be more careful:
X x + y h x i
− 3
b3
b
1/3
b6(x+y)
h i
X x + y h x i
X
x+y
x
=
−
+
−
b3
b3
b3
b3
1/3
b6c−1
c−1
0 y
0 y<b6(x+y)
(
)
x
x
y y max
R 3 , B, 3 +
max
R 3 , B, 3
log x.
1/3
b
B
b
B
B6c−1
c−1
0 y
0 y<B6(x+y)
Lemma 2.1(ii) with k = 6 for the first sum and Lemma 2.2 with k = 3 for the second
yield:
X x + y h x i
− 3
x−1/5 y 6/5 + y 4/5 + x1/7 + x1/21 y 2/3 log x
3
b
b
1/3
b6(x+y)
and one easily checks that
x−1/5 y 6/5 + y 4/5 x1/21 y 2/3
if y 6 x1/3 . The proof of the theorem is complete.
R EFERENCES
[1] M. FILASETA AND O. TRIFONOV, The distribution of fractional parts with applications to gap
results in Number Theory, Proc. London Math. Soc., 73 (1996), 241–278.
[2] A. HILDEBRAND, Multiplicative functions in short intervals, Can. J. Math., 3 (1987), 646–672.
[3] M.N. HUXLEY AND P. SARGOS, Points entiers au voisinage d’une courbe plane de classe C n ,
Acta Arith., 69 (1995), 359–366.
J. Inequal. Pure and Appl. Math., 3(5) Art. 70, 2002
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O LIVIER B ORDELLÈS
[4] M.N. HUXLEY AND P. SARGOS, Points entiers au voisinage d’une courbe plane de classe C n II,
les prépublications de l’Institut Elie Cartan de Nancy (1997).
[5] D.S. MITRINOVIĆ AND J. SÁNDOR (in cooperation with B. Crstici), Handbook of Number Theory,
Kluwer Academic Publisher (1996).
[6] P. SHIU, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math., 313
(1980), 161–170.
J. Inequal. Pure and Appl. Math., 3(5) Art. 70, 2002
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