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volume 5, issue 3, article 81, 2004.

Received 27 August, 2003;

accepted 28 April, 2004.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

CORRIGENDUM TO “ON SHORT SUMS OF CERTAIN MULTIPLICATIVE FUNCTIONS”

OLIVIER BORDELLÈS

2 Allée de la Combe 43000 AIGUILHE FRANCE.

EMail:borde43@wanadoo.fr

c

2000Victoria University ISSN (electronic): 1443-5756 118-03

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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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

The author is indebted to the referee for insightful comments.

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 and2 6 y 6 c₀x^1/2 be real numbers. Let f be a multiplicative function satisfying0 6 f(n)6 1for any positive integern and f(p) = 1for any prime numberp.We have asx→+∞:

X

x<n6x+y

f(n) =yP(f) +O_ε x^1/15+εy^2/3 ,

where

P(f) :=Y

p

1− 1

p

1 +

∞

X

l=1

f p^l p^l

! .

Corrigendum to “On Short Sums of Certain Multiplicative

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Olivier Bordellès

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Before giving the proof, we note that ify < x^1/5,thenx^1/15 > y^1/3 so that the expression x^1/15+εy^2/3 in the error term exceeds the main term (as well as the trivial bound ofy+ 1on the sum).

Proof. On page 5 of [1], the sum

S₁ := X

y<d6x+y dsquarefull

|g(d)|

x+y d

−hx d

i

has been bounded by the sum

S₂ := X

b6(x+y)^1/3

X (_b^y3)^1/2<a6(^x+y_b3 )^1/2

(x+y)b⁻³ a²

− xb⁻³

a²

by usingd=a²b³withµ²(b) = 1,andS₂ has been bounded by

_εx^ε max

16B6(x+y)^1/3

Rx

b³, B, y B³

for any (small) positive real number ε,where we definedR(f, N, δ)to be the number of integer points (n, m) verifying n ∈ ]N; 2N]and |f(n)−m| 6 δ.

Unfortunately, the partP

b6y^1/3 ofS₂ cannot be estimated by

max

16B6(x+y)^1/3

Rx

b³, B, y B³

,

Corrigendum to “On Short Sums of Certain Multiplicative

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Olivier Bordellès

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hence we have to proceed differently: for any positive integerr,we set

τ_(r)(n) :=X

d^r|n

1

and recall that

τ_(r)(n)_εn^ε/r for any positive integersn, r.

InS₁,d=a²b³ > yimpliesa > y^1/5 orb > y^1/5.Since

X

y^1/5<a6(x+y)^1/2

X (_a^y2)^1/3<b6(^x+y_a2 )^1/3

(x+y)a⁻² b³

− xa⁻²

b³

6 X

y^1/5<a6(x+y)^1/2

X

x

a2<n6^x+y_a2

τ₍₃₎(n)

and we have the same ifb > y^1/5,then

S₁ 6 X

y^1/5<b6(x+y)^1/3

X

x

b3<n6^x+y

b3

τ₍₂₎(n) + X

y^1/5<a6(x+y)^1/2

X

x

a2<n6^x+y

a2

τ₍₃₎(n)

= X

y^1/5<b6(2y)^1/3

X

x

b3<n6^x+y_b3

τ₍₂₎(n) + X

(2y)^1/3<b6(x+y)^1/3

X

x

b3<n6^x+y_b3

τ₍₂₎(n)

+ X

y^1/5<a6(2y)^1/2

X

x

a2<n6^x+y_a2

τ₍₃₎(n) + X

(2y)^1/2<a6(x+y)^1/2

X

x

a2<n6^x+y_a2

τ₍₃₎(n)

:= Σ₁+ Σ₂+ Σ₃+ Σ₄.

Corrigendum to “On Short Sums of Certain Multiplicative

Functions”

Olivier Bordellès

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• ForΣ₁andΣ₃ we use the trivial bound:

Σ₁+ Σ_{3 ε} x^ε/2 X

y^1/5<b6(2y)^1/3

x+y b³

−hx b³

i

+x^ε/3 X

y^1/5<a6(2y)^1/2

x+y a²

−h x a²

i

_ε x^ε/2y



 X

b>y^1/5

1

b³ + X

a>y^1/5

1 a²



_ε y^4/5x^ε/2.

• ForΣ₂,we use the method of [1] to get

Σ_{2 ε} x^ε x^1/6+y^1/3 .

• ForΣ₄,if we supposey 6c₀x^1/2(wherec₀ >0is sufficiently small),we have using Lemmas 2.1 and 2.2 of [1]:

Σ_{4 ε} x^ε (

max

(2y)^1/2<A6c⁻¹₀ y

Rx a², A, y

A²

+ max

c⁻¹₀ y<A6x^1/2

Rx

a², A, y A²

)

_ε x^ε

(xy)^1/6+x^1/5+x^1/15y^2/3 .

Hence we finally have S_{1 ε} x^ε

x^1/15y^2/3+y^4/5+ (xy)^1/6+x^1/5 +x^1/6+y^1/3 _ε x^ε x^1/15y^2/3+y^4/5

Corrigendum to “On Short Sums of Certain Multiplicative

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ify>x^1/5.Note thaty^4/5 x^1/15y^2/3ify6c₀x^1/2 and that

X

x<n6x+y

f(n)−yP(f)

y x^1/15y^2/3

ify < x^1/5.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]