Journal of Inequalities in Pure and
Applied Mathematics
CORRIGENDUM TO “ON SHORT SUMS OF CERTAIN
MULTIPLICATIVE FUNCTIONS”
volume 5, issue 3, article 81,
2004.
OLIVIER BORDELLÈS
2 Allée de la Combe
43000 AIGUILHE
FRANCE.
EMail: borde43@wanadoo.fr
Received 27 August, 2003;
accepted 28 April, 2004.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
This note is a corrigendum of the main result of the paper ”On short sums of
certain multiplicative functions” (J. Ineq. Pure & Appl. Math., 3(5), Art. 70
(2002)).
2000 Mathematics Subject Classification: 11N37, 11P21
Key words: Corrigendum, Short sums, Integer points
Corrigendum to “On Short
Sums of Certain Multiplicative
Functions”
The author is indebted to the referee for insightful comments.
Olivier Bordellès
The purpose of this note is both to give a corrected statement of the main
result in [1] and to provide the necessary changes to the arguments in that paper
to justify this corrected statement. The result we now assert is the following :
Theorem 1. Let ε, c0 > 0 and 2 6 y 6 c0 x1/2 be real numbers. Let f be a
multiplicative function satisfying 0 6 f (n) 6 1 for any positive integer n and
f (p) = 1 for any prime number p. We have as x → +∞ :
X
f (n) = yP (f ) + Oε x1/15+ε y 2/3 ,
x<n6x+y
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where
P (f ) :=
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Y
p
1
1−
p
∞
X
f pl
1+
pl
l=1
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J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004
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Before giving the proof, we note that if y < x1/5 , then x1/15 > y 1/3 so that
the expression x1/15+ε y 2/3 in the error term exceeds the main term (as well as
the trivial bound of y + 1 on the sum).
Proof. On page 5 of [1], the sum
S1 :=
X
|g (d)|
y<d6x+y
d squarefull
h i
x+y
x
−
d
d
Corrigendum to “On Short
Sums of Certain Multiplicative
Functions”
has been bounded by the sum
S2 :=
X
b6(x+y)1/3
X
1/2
( by3 )
1/2
−3 (x + y) b−3
xb
−
2
a
a2
<a6( x+y
3 )
Olivier Bordellès
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b
by using d = a2 b3 with µ2 (b) = 1, and S2 has been bounded by
x
y ε
ε x
max
R 3 , B, 3
b
B
16B6(x+y)1/3
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for any (small) positive real number ε, where we defined R (f, N, δ) to be the
number of integer points
P(n, m) verifying n ∈ ]N ; 2N ] and |f (n) − m| 6 δ.
Unfortunately, the part b6y1/3 of S2 cannot be estimated by
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y max
R 3 , B, 3 ,
b
B
16B6(x+y)1/3
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hence we have to proceed differently: for any positive integer r, we set
X
τ(r) (n) :=
1
dr |n
and recall that
τ(r) (n) ε nε/r
for any positive integers n, r.
In S1 , d = a2 b3 > y implies a > y 1/5 or b > y 1/5 . Since
−2 X
X
(x + y) a−2
xa
−
3
b
b3
y 1/5 <a6(x+y)1/2 ( y )1/3 <b6( x+y )1/3
a2
a2
X
X
6
Corrigendum to “On Short
Sums of Certain Multiplicative
Functions”
Olivier Bordellès
τ(3) (n)
x+y
y 1/5 <a6(x+y)1/2 ax2 <n6 a2
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1/5
and we have the same if b > y , then
X
X
S1 6
τ(2) (n) +
x+y
y 1/5 <b6(x+y)1/3 bx3 <n6 b3
X
=
X
X
X
y 1/5 <a6(2y)1/2
x
<n6 x+y
a2
a2
:= Σ1 + Σ2 + Σ3 + Σ4 .
X
τ(3) (n)
x+y
y 1/5 <a6(x+y)1/2 ax2 <n6 a2
X
τ(2) (n) +
x+y
y 1/5 <b6(2y)1/3 bx3 <n6 b3
+
X
(2y)
1/3
τ(3) (n) +
X
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τ(2) (n)
x+y
<b6(x+y)1/3 bx3 <n6 b3
X
X
(2y)1/2 <a6(x+y)1/2
x
<n6 x+y
a2
a2
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τ(3) (n)
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J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004
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• For Σ1 and Σ3 we use the trivial bound:
h i
X
x+y
x
ε/2
− 3
Σ1 + Σ3 ε x
3
b
b
y 1/5 <b6(2y)1/3
h i
X
x+y
x
ε/3
+x
− 2
2
a
a
y 1/5 <a6(2y)1/2


X 1
X 1
 ε y 4/5 xε/2 .
ε xε/2 y 
+
3
2
b
a
1/5
1/5
b>y
a>y
• For Σ2 , we use the method of [1] to get
Σ2 ε xε x1/6 + y
Corrigendum to “On Short
Sums of Certain Multiplicative
Functions”
Olivier Bordellès
1/3
.
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• For Σ4 , if we suppose y 6 c0 x (where c0 > 0 is sufficiently small), we
have using Lemmas 2.1 and 2.2 of [1]:
(
)
x
x
y
y
Σ 4 ε x ε
max
R 2 , A, 2 + −1 max
R 2 , A, 2
a
A
a
A
c0 y<A6x1/2
(2y)1/2 <A6c−1
0 y
ε xε (xy)1/6 + x1/5 + x1/15 y 2/3 .
Hence we finally have
S1 ε xε x1/15 y 2/3 + y 4/5 + (xy)1/6 + x1/5 + x1/6 + y 1/3
ε xε x1/15 y 2/3 + y 4/5
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J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004
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if y > x1/5 . Note that y 4/5 x1/15 y 2/3 if y 6 c0 x1/2 and that
X
f (n) − yP (f ) y x1/15 y 2/3
x<n6x+y
if y < x1/5 . This concludes the proof of the theorem.
Corrigendum to “On Short
Sums of Certain Multiplicative
Functions”
Olivier Bordellès
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References
[1] O. BORDELLÈS, On short sums of certain multiplicative functions, J.
Inequal. Pure and Appl. Math., 3(5) (2002), Art. 70. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=222]
Corrigendum to “On Short
Sums of Certain Multiplicative
Functions”
Olivier Bordellès
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J. Ineq. Pure and Appl. Math. 5(3) Art. 81, 2004
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