Int. Journal of Math. Analysis, Vol. 7, 2013, no. 3, 123 - 153
Average of Some Multiplicative Functions
on the Set of Integers without Large Prime Factor
Mohamed Saber DAOUD
Département de Mathématiques, Faculté des Sciences de Tunis
Université de Tunis EL MANAR
Campus Universitaire, 2092 Tunis, Tunisie
med.saber.daoud@gmail.com
Afef HIDRI
Département de Mathématiques, Faculté des Sciences de Tunis
Université de Tunis EL MANAR
Campus Universitaire, 2092 Tunis, Tunisie
afefhidri@yahoo.fr
Abstract
Let λ > 0 , δ ∈]0, 1[ and f (n) be a multiplicative function satisfying essentially
for a large x ; f (p)≤x log f (p) = λx+ O(x(log x)−δ ). In the present work, we estab
1
and f (n)≤x;P (n)≤y 1
lish an asymptotic formula for the two sums f (n)≤x;P (n)≤y f (n)
, valid in the domain 1 ≤ cx ≤
1
δ
C0 (log f (y)) ,
for a suitable constants c and C0 .
Mathematics Subject Classification: 11N37 (11N25)
Keywords: asymptotic formula; multiplicative function; difference-differential equation
124
1
M. S. DAOUD and A. HIDRI
Introduction
Let P (n) denote the largest prime dividor of a positive integer n, with P (1) = 1. Also,
throughout the paper, the letters p, p1 and p2 denote a prime number.
For x ≥ 1 and y > 1, we introduce the quantity :
Ff (x, y) =
f (n)≤x
1
, Sf (x, y) =
f (n)
P (n)≤y
1, ψf (x, y) =
f (n)≤x
n≤x
P (n)≤y
P (n)≤y
f (n).
Obviously, Sf (x, y), Ff (x, y), and ψf (x, y) are a natural generalization of the well-know
function, usually denoted by ψ(x, y), which is the number of integers ≤ x and free of
prime factors > y. The later has been the object of a number of articles in the last three
decades (see for example: [2], [1], [5], [10], [4] et [7]).
In [6], Naimi gives an asymptotic formula for Sf (x, y), for a multiplicative function
such that f (p) = λp , (λ > 1):
y
log x + log(v + 1)
)
Sf (x, y) = F.x.(log( ))λ−1 ρλ (v){1 + Oε ( 2
λ
log( λy )
(1)
uniformly in the domain Hε,λ,c (x0 ), defined by
x ≥ x0 , exp(log2 cx)5/3+ε ≤
with ε > 0, c =
1
inf{f (n);n≥1}
, v =
log cx
log y/λ
y
≤ cx
λ
and λ to be the continuous solution of the
differential difference equation :
uλ (u) = (λ − 1)λ (u) − λλ (u − 1), u > 1, λ > 0 ,
with initial conditions
⎧
⎨ 0 si u ≤ 0
λ (u) =
⎩ uλ−1 si 0 < u ≤ 1
Γ(λ)
The function λ , generalizes Dickman’s function . Many paper have been written on
proprieties of this function ( see for example [8] et [3] ).
125
Average of some multiplicative functions
In his paper [9], J.M.Song established an asymptotic formula for ψf (x, y) where f is
non-negative multiplicative arithmetic function satisfying:
There exists δ ∈]0, 1[, λ > 0 and b > 0 such as
f (p)
p
p≤z
log p = λ log z + O((log z)1−δ ), z ≥ 2
and
f (pk )
k≥2
pk
log pk ≤ b.
The aim of this paper is to give estimates of the two sums Ff (x, y) and Sf (x, y) for a class
of arithmetic function where the principals conditions are inspired from [9].
Let ξλ,δ denote the class of non-negative multiplicative function f satisfying the following conditions :
(Ω1 )
log f (p) = λz + O(z(log z)−δ ); λ ≥ 1, δ ∈]0, 1[
f (p)≤z
(Ω2 )
log f (pk )
f (pk )
(log z)−δ
f (pk )≤z
.
(p,k)≥2
p1 < p2 ⇒ p1 f (p1 ) < f (p2 )
(Ω3 )
Hypothesis (Ω1 ) is often replaced by the weaker hypothesis
(Ω∗1 )
log(f (p))
= λ log z + O((log z)1−δ ) ,
f (p)
f (p)≤z
obtained from (Ω1 ) using Abel summation.
Before starting our results, we introduce some notations.
For f ∈ ξλ,δ , we note v =
log cx
log f (y)
;(in this article, we confuse y and the largest prime
1
;(c
inf {f (n),n≥1}
number less than y), c =
≥ 1) and throughout the paper, O and depends
at most than λ.
Let jλ the continuous solution of the difference-differential equation
ujλ (u) = kjλ (u) − λjλ (u − 1), u > 1, λ > 0 ,
126
M. S. DAOUD and A. HIDRI
with initial conditions :
⎧
⎨ 0
jλ (u) =
⎩
si u ≤ 0
uλ
Γ(λ+1)
si 0 < u ≤ 1
The function jλ , verifying jλ (v) = ρλ (v), was studied in [9]. We gives it’s principals
properties in Lemma 1.
Our main result is the following:
Theorem 1:
Let f ∈ ξλ,δ . There exist a constant A(f ) such as for all sufficiently large x:
Ff (x, y) = A(f ).(log f (y))λ.jλ (v).{ 1 + O(
log v
) }
(inf(log f (y), log x))δ
uniformly in
(Hf,δ,C0 )
0 ≤ v ≤ exp(
1
(log(f (y))δ ))
C0
for a suitable constant C0 .
The following theorem is the generalization of (1):
Theorem 2:
Let f ∈ ξλ,δ . For all sufficiently large x
Sf (x, y) = A(f )x(log f (y))λ−1ρλ (v){1 + O(
log v
)}
(inf(log f (y), log x))δ
uniformly in (Hf,δ,C0 ), where A(f ) is the constant of theorem 1.
2
Preliminary lemmas
Lemma 1 : Properties of jλ :
127
Average of some multiplicative functions
1)jk (u) is a strictly increasing function in [0, +∞[
(2.1.1)
2) jk (u) converges to 1 as u → +∞
3) jλ (u) ≥
1
2
(2.1.2)
, for u > λ
(2.1.3)
4)For v ≥ 0 and a ∈]0, v] :
jλ (v) jλ (a)
− λ = −λ.
vλ
a
v
a
5)For v ∈]1, 2]:
1
.[v λ − λ.
jλ (v) =
Γ(λ + 1)
6)For v ≥ 1
1
− λ.
jλ (v) = v .[
Γ(λ + 1)
λ
jλ (x − 1)
dx
xλ+1
v
(v − h)λ
dh]
h
(2.1.5)
jλ (u − 1)
du]
uλ+1
(2.1.6)
1
1
v
(2.1.4)
7) Forv ≥ 1
λ
jλ (v) =
v
v
1
jλ (t) dt +
v
v−1
v
jλ (t) dt +
1
1
.
v.(λ + 1).Γ(λ + 1)
(2.1.7)
Proof :
1), 2), 3) and 7) :see [11].
4)We have (x−λ jλ (x)) =
1
[−λjλ (x)
xλ+1
+ xjλ (x)] =
1
.(−λjλ (x
xλ+1
− 1)).
When we integrate the two sides of this equality with respect to x from a to v, we
obtain the result.
5)The proof is deduced from 4) in the particular case v ∈]1, 2] and a = 1. Indeed
v
λ
1
jλ (v)
(x − 1)λ
=
−
.
−
dx
vλ
Γ(λ + 1)
Γ(λ + 1) 1
xλ+1
v
v
1
1
λ
[v − λ.
(v − )λ dx].
=⇒ jk (v) =
Γ(λ + 1)
x
1 x
With the change of variable h =
v
x
we obtain :
1
jλ (v) =
.[v λ − λ.
Γ(λ + 1)
1
v
(v − h)λ
dh
],
h
(v ∈]1, 2]).
128
M. S. DAOUD and A. HIDRI
This proves (2.1.5)
6)Taking a = 1 in the equation (2.1.4).
Lemma 2:
Let f ∈ ξλ,δ . We have
1)
L(y, t) = λ log2 t − λ log2 (f (y)) + O((log f (y))−δ ) , (t > 0)
(2.2.1)
log f (p) x 1 log f (p)
d(
) (log x)1−δ
=
2
(f (p))
(f (p)
1/c t
(2.2.2)
2)
f (p)≤x
f (p)≤t
3)
f (pk )≤x
1
(log x)−δ .
f (pk )
(2.2.3)
k≥2
Proof :
1)by Abel summation based on (Ω∗1 ) we deduced that
L(y, t) =
f (y)<f (p)≤t
1
=
f (p)
t
f (y)
log f (p)
1
.d(
)
log x
f (p)
f (p)≤x
= λ log2 t − λ log2 (f (y)) + O((log f (y))−δ ).
2)Using Abel summation based on (Ω∗1 ), we obtain (2.2.2)
3)We obtain (2.2.3) by (Ω2 ).
Lemma 3 :
Suppose f ∈ ξλ,δ .Then for v ∈ [1, λ + 2] we have
f (y)<f (p)≤cx
Proof :
v
x λ
1
1
(v − h)λ
λ
.(log(
) = λ.(log f (y)) .[
dh + O(
)].
f (p)
f (p)
h
(log f (y))δ
1
(2.3)
129
Average of some multiplicative functions
Let M denote the sum on the left hand side of the lemma. By Abel summation, we
get:
cx
x
(log( ))λ .d(
M=
t
f (y)
f (y)<f (p)≤t
x
t=cx
= [(log( ))λ .L(y, t)]t=f
(y) + λ.
t
1
)=
f (p)
cx
f (y)
cx
x
(log( ))λ .d(L(y, t))
t
f (y)
(log( xt ))λ−1
L(y, t) dt = A + B
t
=⇒ M = A + B.
The quantity A is equal to
(log(
x
x λ
)) .[λ log2 cx − λ log2 f (y) + O((log x)−δ )] − (log(
))λ .[λ log2 f (y)
cx
f (y)
−λ log2 f (y) + O((log x)−δ )].
And B is equal to
x
−[(log( ))λ .λ. log2 (t)]t=cx
t=f (y) +
t
cx
f (y)
λ.
(log( xt ))λ
dt
t log t
x
λ−δ
).
+λ. log2 f (y)[(log( ))λ ]t=cx
t=f (y) + O((log(f (y)))
t
After simplification, we get
(log( xt ))λ
dt + O((log f (y))λ−δ ).
λ.
M=
t
log
t
f (y)
Let G denote the integral
cx
cx
f (y)
λ.
(log( xt ))λ
dt
t log t
(log( xt ))λ
dt
t log t
f (y)
cx
1
log ct λ
1
λ
)].
) .
dt.
(v −
= λ.(log f (y)) [1 + O(
log f (y)
log f (y) t log ct
f (y)
G=
With the change of variable h =
cx
λ.
log ct
,we
log f (y)
obtain
1
)].
G = λ.(log f (y)) [1 + O(
log f (y)
λ
log c
v+ log
f (y)
log c
1+ log
f (y)
(v − h)λ
dh.
h
130
M. S. DAOUD and A. HIDRI
Let
U=
We remarks that
U=
v
1
(v − h)λ
dh −
h
v
=
1
log c
v+ log
f (y)
log c
1+ log
f (y)
log c
1+ log
f (y)
1
(v − h)λ
dh −
h
(v − h)λ
dh.
h
(v − h)λ
dh −
h
log c
1+ log
f (y)
1
It’s clearly that U1 and U2 are at most v
log c
v+ log
f (y)
(v − h)λ
dh
h
(v − h)λ
dh − U1 − U2 .
h
1
,
log f (y)
then
v
1
1
(v − h)λ
G = λ.(log f (y)) [1 + O(
)].[
dh + O(
)]
log f (y)
h
log f (y)
1
v
(v − h)λ
1
λ
= λ.(log f (y)) .[
dh + O(
)].
h
log f (y)
1
λ
This proves the lemma
Lemma 4:
Suppose f ∈ ξλ,δ and v ∈]0, λ + 2], then
Ff (x, y) = Ff (x) −
f (y)<f (p)≤cx
where Ff (x) = Ff (x, x) =
1
x
.Ff (
, p) + O((inf(log f (y), log x))λ−δ )
f (p)
f (p)
1
f (n)≤x f (n) .
Proof
Ff (x, y) = Ff (x) −
f (n)≤x
1
= Ff (x) − S0 .
f (n)
∃p/n;p>y
When we rearrange the sum S0 following the largest prime divisor of n, we obtain
S0 =
p>y
f (n)≤x
P (n)=p
1
f (n)
(2.4)
131
Average of some multiplicative functions
=
p>y
1
+
f (n) p>y
f (n)≤x
P (n)=p,pn
=
1
f (p)
p>y
f (p)≤cx
f (y)<f (p)≤cx
+
x
f (m)≤ f (p)
1
f (pk )
p>y
Ff (cx)
f (y)
cx
f (y)
1
+
f (m)
1
f (pk )
p>y
f (m)≤
f (pk )≤cx;k≥2
f (m)≤
f (pk )≤cx;k≥2
We verify that R1 ∃k≥2;pk n
1
x
.Ff (
, p) −
f (p)
f (p)
1
f (n)
f (n)≤x;P (n)=p
P (m)<p
=
x
f (pk )
f (y)<f (p)≤cx
x
f (pk )
1
f (m)
P (m)<p
1
f (p)
x
f (m)≤ f (p)
1
f (m)
P (m)=p
1
= Tp + R1 + R2 .
f (m)
P (m)<p
(log x)λ−δ . And by Abel summation, R2 is at most
(log( xt ))λ
d(
log t
f (pk )≤t ; k≥2
log f (pk )
) (log x)λ−δ .
f (pk )
This proves (2.4)
Lemma 5:
Let f ∈ ξλ,δ . For v ∈ [1, λ + 2]
f (y)<f (p)≤cx
(log f (p))λ jλ (vp − 1)
= λ.v λ (log f (y))λ.
f (p)
v
1
jλ (u − 1)
du + O((log f (y))λ−δ );
uλ+1
(2.5)
où vp =
log cx
.
log f (p)
Proof : Let I denote the sum on the left side of (2.5) and a double integration by parts
applied to I gives
I =
cx
f (y)
= λ.
(log t)λ .jλ (
cx
f (y)
jλ (
log cx
− 1).d(
log t
f (y)<f (p)≤t
1
)
f (p)
dt
log cx
− 1)(log t)λ
+ O((log f (y))λ−δ ).
log t
t log t
132
M. S. DAOUD and A. HIDRI
With the change of variable u =
log cx
,
log t
we obtain (2.5).
Lemma 6:
(log cx)Ff (x, y) ≤ (log c).Ff (x, y) +
x
1
c
Ff (t, y)
dt +
t
Ff (
1≤f (pk )≤cx
log f (pk )
x
,
y).
;
f (pk )
f (pk )
p≤y;k≥1
(2.6)
(v ≥ 1 , f ∈ ξ).
Proof:
This inequality can be easily established by evaluating the sum S =
log(cf (n))
,
f (n)
f (n)≤x
P (n)≤y
in two different ways: On the one hand, we have :
x
S=
log ct d(Ff (t, y)) = (log cx).Ff (x, y) −
1
c
On the other hand, partial summation yields
S = (log c).Ff (x, y) +
f (n)≤x
1
f (n)
x
1
c
Ff (t, y)
dt (∗).
t
log f (pk )
pk //n ;k≥1
P (n)≤y
=⇒ S = (log c).Ff (x, y) +
log f (pk )
f (pk )
p≤y
f (pk )≤cx ;k≥1
=⇒ S ≤ (log c).Ff (x, y) +
f (m)≤
x
f (pk )
1
f (m)
P (m)≤y ;(m,p)=1
x
log f (pk )
.Ff (
, y) (∗∗)
k
f (p )
f (pk )
p≤y
1≤f (pk )≤cx
;k≥1
and the result follows on equating the two expressions (*) and (**).
Lemma 7:
Suppose f ∈ ξ and for θ ∈]0, 1[; we have:
θ
log f (p) log f (p)
).
= λ log f (y) .
jλ (v −
jλ (v − u)du + O((log f (y))1−δ ).
log f (y)
f (p)
0
θ
1≤f (p)≤(f(y))
(2.7)
133
Average of some multiplicative functions
Proof:
We consider s(z) =
f (p)≤z
log f (p)
f (p)
and r(z) = s(z) − λ log z.
One deduces from (Ω∗1 ) that there exists a constant A > 0 such that :
|r(z)| ≤ A. (log z)1−δ .
Denote the left -hand side of the above formula by S. Partial summation yields
S=
= λ.
(f (y))θ
1
(f (y)θ
log t
) d(s(t))
log f (y)
jλ (v −
1
dt
log t
) +
jλ (v −
log f (y) t
(f (y))θ
jλ (v −
1
log t
) d[s(t) − λ log t]
log f (y)
= M + R.
With the change of variable :u =
log t
,
log f (y)
we obtain
M = λ log f (y) .
0
θ
jλ (v − u) du.
The term R is at most of order
jλ (v − θ). r((f (y))θ ) −
(f (y))θ
1
r(t).
d
log t
[jλ (v −
)] dt
dt
log f (y)
=⇒ jλ (v − θ). r((f (y))θ ) + (log f (y))1−δ .
1
(f (y))θ
jλ (v −
log t
) dt
log f (y)
jλ (v − θ). r((f (y))θ ) + (log f (y))1−δ (jλ (v − θ) − jλ (v))
(log f (y))1−δ .
This complete the proof of Lemma.
3
Proof of theorem 1
Proposition 0 :
Let f ∈ ξλ,δ .Then for all sufficiently large x :
Ff (x) = C(f ).(log x)λ + O((log x)λ−δ ); λ ≥ 1, δ ∈]0, 1[
(3.1)
134
M. S. DAOUD and A. HIDRI
where C(f ) =
A(f )
and A(f ) defined in theorem 1.
Γ(λ + 1)
Proof of proposition 0:
On the one hand, we have
log(f (n))
1 log f (pk ) =
=
f (n)
f (n) k
f (n)≤x
f (n)≤x
p //n
f (m)f(pk )≤x
log f (pk )
= D.
f (pk )f (m)
(P.1)
(m,p)=1;k≥1
On the other hand, partial summation yield
x
log(f (n)) x
Ff (t)
=
dt.
log t d(Ff (t)) = (log x)Ff (x) −
f (n)
t
1/c
1/c
(P.2)
f (n)≤x
The sum on the right of (P.1) is equal to
f (m)f(p)≤x
log f (p)
+
f (m)f (p)
f (m)f(pk )≤x
(m,p)=1
log f (pk )
= D 1 + D2
f (m)f (pk )
(m,p)=1;k≥2
after separating terms corresponding to k = 1 and k ≥ 2. By (Ω2 ), the sum D2 is at most
f (m)≤x
1 log(f (pk ))
Ff (x).
f (m) p,k≥2 f (pk )
The quantity D1 is equal to
f (m)≤x
1
f (m)
x
f (p)≤ f (m)
log f (p)
−
f (p)
f (l)f (p)f (pk )≤x
log f (p)
= B1 + B2
f (l)f (p)f (pk )
(p,l)=1;k≥1
Define
T (x) =
x
1/c
x
log( f (m)
)
Ff (t)
dt =
.
t
f (m)
(P.3)
f (m)≤x
So that, by (Ω∗1 )
B1 =
f (m)≤x
x
1
{λ log(
) + O((log x)1−δ )} = λ.T (x) + O(Ff (x)(log x)1−δ ).
f (m)
f (m)
135
Average of some multiplicative functions
Using (2.2.2) and (2.2.3), the sum B2 is at most
f (l)(f(p))2 ≤x
log f (p)
+
f (l)(f (p))2
f (l)(f(p))(f(pk ))≤x
log f (p)
f (l)f (p)(f (pk ))
k≥2
f (l)≤x
1
f (l)
√
f (p)≤ cx
1
log f (p)
+
(f (p))2
f (l)
f (l)≤x
(f (p))(f(pk ))≤cx
log f (p)
Ff (x)(log x)1−δ .
f (p)(f (pk ))
k≥2
Finally, after appeal to (P.1) , (P.2) and (P.3), we conclude that
(log x)Ff (x) = T (x) + O(Ff (x)) + [λ.T (x) + O(Ff (x)(log x)1−δ )].
And, for
ε(x) (log x)−δ
(P.4)
we have
Ff (x) =
λ+1
T (x) + Ff (x)ε(x).
log x
If x is a large enough, say x ≥ x0 , then
|ε(x)| ≤ 1/2, x ≥ x0
(P.5)
and we have the useful expression
Ff (x) =
1
λ+1
.
.T (x), x ≥ x0 .
1 − ε(x) log x
Let
E(x) = log(
(λ + 1)T (x)
)
(log x)λ+1
and note that, by (P.3) and (P.6)
E (x) =
Hence E0 =
∞
x
1
(λ + 1)ε(x)
, x ≥ x0 .
x log x(1 − ε(x))
(log x)1+δ
E (x)dx converge absolutely, and therefore
∞
x
∞
E (z)dz =
E (z)dz −
E (z)dz = E0 − (E(x) − E(1+ )).
1+
1+
1+
(P.6)
136
M. S. DAOUD and A. HIDRI
On writing C(f ) = exp(E0 + E(1+ ))
(λ + 1).T (x)
exp(E(x)) =
= C(f ). exp(−
(log x)(λ+1)
∞
x
E (z)dz) = C(f )[1 + O((log x)−δ )].
When we substitute this in (P.6) and use (P.4), we obtain
Ff (x) = C(f ).(log x)λ + O((log x)λ−δ ); λ ≥ 1, δ ∈]0, 1[.
(P.7)
To determinate C(f ), we agree as follows. For s > 1 and u ∈ R, by Abel summation,
we have
f (n)≤u
1
=
(f (n))s
u
1/c
1−s
t
d(Ff (t)) = Ff (u)u
1−s
+ (s − 1).
u
1/c
Ff (x)
dx.
xs
(P.8)
Using (P.7)
Ff (u)u1−s = u1−s [C(f ).(log u)λ + O((log u)λ−δ )]
then
lim Ff (u)u1−s = 0.
(P.9)
u−>∞
The quantity on the right of (P.8) is equal ( by (P.7) )to
+∞
+∞
C(f )(log x)λ
C(f )(log x)λ−δ
(s−1)[
dx+O(
dx+1)] = C(f ).Γ(λ+1)(s−1)−λ+O(s−1).
s
s
x
x
1
1/c
When we substitute this in (P.8) and using (P.9), we obtain
f (n)≥1/c
So that
C(f ) =
1
Γ(λ + 1)
=
C(f
).
+ O(s − 1).
f (n)s
(s − 1)λ
1
. lim+ [(s − 1)λ .
Γ(λ + 1) s−>1
f (n)≥1/c
1
].
f (n)s
On the other hand lims→1+ [(s − 1)ζ(s)] = 1, hence
C(f ) =
1
. lim+ [ ( (1 − p−s )−λ ).(
Γ(λ + 1) s→1
p
f (n)≥1/c
This proves the proposition.
1
) ].
f (n)s
137
Average of some multiplicative functions
Theorem 1 is obtained by combining the two following propositions.
Proposition 1 :
Let f ∈ ξλ,δ . For v ∈]0, λ + 2], we have
Ff (x, y) = A(f )(log(f (y))λjλ (v){1 + O((inf(log f (y), log x))−δ )}.
(3.2)
Proposition 2 :
Let f ∈ ξλ,δ .For λ + 2 ≤ v ≤ exp( C10 (log(f (y))δ ))
Ff (x, y) = A(f ).(log f (y))λ.jλ (v).[ 1 + O(
log v
) ].
(log f (y))δ
(3.3)
Proof of proposition 1:
We proceed inductively.we will proves that
For y ≥ y0 , for all k ∈ [1, λ + 2] and for all v ∈]k − 1, k[, we have :
Ff (x, y) = A(f )(log(f (y))λjλ (v){1 + O((inf(log f (y), log x))−δ )}.
(R)
Let Tp denote the sum on the right hand side of (2.4) .
• k = 1, v ∈]0, 1] ⇐⇒ 1 < cx ≤ f (y):
We verify that Tp = 0, hence the expression (2.4) became :
Ff (x, y) = Ff (x) + O((log x)λ−δ ) = Ff (x)
(P.1.1).
log x λ
) = 1 + O( log1 x ), then we deduce from (3.1) that
And we remarks that ( log
cx
Ff (x, y) = A(f )
vλ
(log f (y))λ[1 + O((log x)−δ )] (y ≥ y0 )
Γ(λ + 1)
which establishes the equality (R) for k = 1, since jk (v) =
vλ
Γ(λ+1)
(P.1.2)
in ]0, 1].
• k = 2, v ∈]1, 2] ⇐⇒ f (y) < cx ≤ (f (y))2:
x
In this case, we remarks that for f (y) < f (p), the Ff ( f (p)
, p) verify (P.1.1), hence
Tp = C(f ).
f (y)<f (p)≤cx
x
1
.(log(
))λ + O(
f (p)
f (p)
f (y)<f (p)≤cx
x
1
.(log(
))λ−δ ) = M + R.
f (p)
f (p)
Using (2.2.1), the error term bellows is at most
f (y)<f (p)≤cx
x
1
.(log(
))λ−δ (log f (y))λ−δ .
f (p)
f (p)
f (y)<f (p)≤cx
1
(log f (y))λ−δ
f (p)
138
M. S. DAOUD and A. HIDRI
and after appeal to lemma 3, the sum on (P.1.2) is equal to
v
(v − h)λ
1
λ
)].
C(f ).λ.(log f (y)) .[
dh + O(
h
(log f (y))δ
1
Hence
Tp = C(f ).M + R = C(f ).λ.(log f (y)) .[
v
λ
1
(v − h)λ
1
)].
dh + O(
h
(log f (y))δ
When we substitute the expression of Tp in (2.4), we have
v
(v − h)λ
λ
dh + O((log(f (y))λ−δ )
Ff (x, y) = Ff (x) − C(f ).λ.(log f (y)) .
h
1
and using (2.1.5) and proposition 0, we obtain:
Ff (x, y) = A(f )(log f (y))λjλ (v)[1 + O((log f (y))−δ )].
This proves (R) for k = 2. Let k ∈ [2, λ + 1]; assume that (R) is verified for all integers
k0 ∈ [1, k] and we have to prove that (R) is also true for k + 1. For this, let v ∈]k, k + 1].
We remarks that for f (p) > f (y), vp =
log( fcx
)
(p)
log f (p)
=
log cx
log f (p)
− 1 ≤ v − 1 ≤ k,hence, when
x
, p), we obtain
we applying the induction hypothesis to the term Ff ( f (p)
Ff (
x
, p) = A(f )(log f (p))λ jλ (vp ).[1 + O((log f (y))−δ )].
f (p)
So
Tp = A(f ).[
f (y)<f (p)≤cx
(log f (p))λ jλ (vp )
+ O(
f (p)
f (y)<f (p)≤cx
(log f (p))λ jλ (vp )
(log f (y))−δ ) ]
f (p)
= A(f ).[I + R].
On the one hand, using lemma 5, we have
v
jλ (u − 1)
λ
λ
du + O((log f (y))λ−δ ).
I = λ.v (log f (y)) .
λ+1
u
1
On the other hand, by (2.2.1) and since jλ (.) 1, we obtain:
R (log f (y))λ−δ ).
f (y)<f (p)≤cx
1
(log f (y))λ−δ ).λ log(v) (log f (y))λ−δ ).
f (p)
139
Average of some multiplicative functions
Hence, the expression (2.4) is
λ
λ
Ff (x, y) = Ff (x) − λ.v (log f (y)) .
v
1
jλ (u − 1)
du + O((log f (y))λ−δ ).
uλ+1
(P.1.3)
Finally, using proposition 0 and(2.1.6), the expression (P.1.3) is equal to
Ff (x, y) = A(f )(log f (y))λjλ (v)[1 + O((log f (y))−δ )]
and (R) is verified of k + 1. this proves the proposition .
proof of proposition 2 :
We deduce from proposition 1 that of v ∈ [0, λ + 2]
Ff (x, y) = A(f )(log f (y))λjλ (v)[1 + O((log f (y))−δ )].
(P.2.1)
For v ≥ λ, define Δ(v, y) by the following formula
Ff (x, y) = A(f )(log f (y))λjλ (v)[1 + Δ(v, y)]
Let
(P.2.2)
Δ (v, y) = supλ+1≤v ≤v Δ(v , y)
∗
then we have by (P.2.1)
Δ∗ (v, y) (log f (y))−δ , λ < v ≤ λ + 2.
(P.2.3)
Our goal here is to prove
Δ∗ (v, y) log v
(log f (y))δ
(P.2.4)
uniformly for v ≥ λ + 2 and for all sufficiently large y to prove the proposition. First, we
return to the inequality in Lemma 6.
By (P.2.2), for v ≥ λ + 2, we have
(log cx){A(f )(log f (y))λjλ (v)[1 + Δ(v, y)]} ≤ (log c)A(f )(log f (y))λjλ (v)[1 + Δ(v, y)]
x
log f (pk )
log f (pk )
Ff (t, y)
log f (pk )
dt+
).[1+Δ(v−
, y)].
.
A(f )(log f (y))λjλ (v−
+
k)
1
t
log
f
(y)
log
f
(y)
f
(p
c
k
1≤f (p )≤cx
p≤y ; k≥1
(P.2.5)
140
M. S. DAOUD and A. HIDRI
The sum on the right of (P.2.5) is equal, with the notation vp =
log f (pk )
log f (y)
log f (p)
log f (y)
and vpk =
; (k ≥ 2), to
log f (pk )
log f (pk )
log f (pk )
).[1 + Δ(v −
, y)].
log f (y)
log f (y)
f (pk )
A(f )(log f (y))λjλ (v −
1≤f (pk )≤cx
p≤y ; k≥1
=
A(f )(log f (y))λjλ (v − vp ).[1 + Δ(v − vp )].
1≤f (p)≤cx
log f (p)
f (p)
p≤y
+
A(f )(log f (y))λjλ (v − vpk ).[1 + Δ(v − vpk )].
1≤f (pk )≤cx
log f (pk )
.
f (pk )
p≤y ; k≥2
We consider the integral in (P.2.5), namely
x
1
c
Ff (t, y)
dt =
t
f (y)
c
1
c
Ff (t)
dt +
t
x
f (y)
c
Ff (t, y)
dt.
t
By proposition 0, the first integral on the right is
C(f )
(log f (y))λ+1.[1 + O((log f (y))−δ )].
λ+1
Using proposition 1 and the changement of variables u =
integral that corresponds to
f (y)
c
A(f )(log f (y))
≤t≤
λ+1
(f (y))λ+2
λ+2
1
c
log ct
,
log f (y)
the portion of the second
is equal to
jλ (u)du + O((log f (y))λ+1−δ )
and the remaining portion contribues, with the notation in (P.2.2)
v
λ+1
jλ (u).[1 + Δ(u, y)]du.
= A(f )(log f (y))
λ+2
Hence
λ+2
v
x
C(f )
Ff (t, y)
λ+1
λ+1
dt =
(log f (y)) +A(f )(log f (y)) .[
jλ (t)dt+
jλ (t) . Δ(t, y) dt]
1
t
λ+1
1
λ+2
c
+O((log f (y))λ+1−δ ) ; C(f ) =
A(f )
.
Γ(λ + 1)
141
Average of some multiplicative functions
And, we deduced from the above, that the inequality (P.2.5) is
1+Δ(v, y) ≤ [1+Δ(v, y)]
+
1
log c
+
log cx jλ (v) log cx
1
jλ (v) log cx
1≤f (p)≤f (y)
log f (p)
jλ (v−vp ).[1+Δ(v−vp , y)]
f (p)
log f (pk )
jλ (v − vpk ).[1 + Δ(v − vpk , y)]
f (pk )
1≤f (pk )≤cx
p≤y ; k≥2
log f (y)
log f (y)
+
.
+
(λ + 1)Γ(λ + 1)(log cx)jλ (v) (log cx)jλ (v)
+O(
v
1
log f (y)
.
jλ (t) dt+
(log cx)jλ (v)
v
λ+2
(log f (y))1−δ
).
jλ (v) log cx
jλ (t) Δ(t, y) dt
(P.2.6)
The second sum on the right of (P.2.6) is at most
1 + Δ∗ (v, y)
log cx
by (Ω2 ). And using Lemma 7 (θ = 1), the first sum is equal to
λ(log f (y))
.
jλ (v). log cx
+
We remarks that
1
jλ (v) log cx
1
0
1
jλ (v − t) dt + O(
0
1≤f (p)≤f (y)
jλ (v − t) dt =
v
(log f (y))1−δ
)
jλ (v) log cx
log f (p)
jλ (v − vp ).Δ(v − vp , y).
f (p)
j (t) dt,
v−1 λ
hence, with (2.1.7), the inequality (P.2.6)
simplifies to
1
Δ(v, y) ≤
jλ (v) log cx
1≤f (p)≤f (y)
log f (p)
1
jλ (v−vp ).Δ(v−vp , y)+
.
f (p)
v jλ (v)
+O(
1 + Δ∗ (v, y)
).
v.(log f (y))δ
v
λ+2
jλ (t).Δ(t, y) dt
(P.2.7)
By vertue of Lemma 7 (θ = 1/2), the sum of the right of (P.2.7) that correspond to
1 ≤ f (p) ≤ (f (y))1/2 contribute at most
Δ∗ (v, y)
.[λ
v jλ (v)
0
1
2
jλ (v − t) dt + O((log f (y))−δ )]
142
M. S. DAOUD and A. HIDRI
(since 0 <
log f (p)
log f (y)
≤ 12 ) and the remaining term contribute at most
Δ∗ (v − 12 , y)
.[λ.
v jλ (v)
(since
1
2
<
log f (p)
log f (y)
1
1
2
jλ (v − t) dt + O((log f (y))−δ )]
≤ 1).
We consider the integral in (P.2.7) . we have
1
.
v.jλ (v)
v
v−1
1
jλ (t).Δ(t, y) dt ≤
jλ (t)Δ(t, y) dt +
jλ (t)Δ(t, y) dt]
.[
v.jλ (v) 1
λ+2
v−1
Δ∗ (v, y) v
Δ∗ (v − 1, y) v−1
jλ (t) +
jλ (t) dt
≤
v.jλ (v)
v.jλ (v) v−1
1
Δ∗ (v − 12 , y) v−1
Δ∗ (v, y) v
jλ (t) +
jλ (t) dt
≤
v.jλ (v)
v.jλ (v) v−1
1
v
and thus
1
|Δ(v, y)| ≤ Δ (v, y).[
v.jλ(v)
∗
v
1
1
jλ (t)dt+α(v)]+Δ (v− , y).[
2
v.jλ (v)
v−1
∗
+O(
1 + Δ∗ (v, y)
).
v(log f (y))δ
where
λ
α(v) =
v.jλ (v)
and
λ
β(v) =
v.jλ (v)
1
2
0
jλ (v − t)dt
1
1
2
jλ (v − t)dt.
We note that
α(v), β(v) ≤
λ
, v>1
2v
by the monotonicity of jλ(.) .
Introduce
1
α1 (v) =
v.jλ (v)
v
v−1
jλ (t)dt + α(v)
1
v−1
jλ (t)dt+β(v)]
(P.2.8)
143
Average of some multiplicative functions
and note, by (2.1.7), that
v
v
v
1
1
λ
jλ (t)dt +
jλ (t)dt] − [
jλ (t)dt + α(v)]
= [
v.jλ (v) v−1
v.jλ (v) 1
v.jλ (v) v−1
v− 1
v−1
2
λ
1
=
jλ (t)dt +
jλ (t)dt
v.jλ (v) v−1
v.jλ (v) 1
v−1
1
jλ (t)dt ≤ 1 − α1 (v).
= β(v) +
v.jλ (v) 1
Hence, (P.2.8) simplifies to
1
1 + Δ∗ (v, y)
)
|Δ(v, y)| ≤ Δ∗ (v, y) α1(v)+Δ∗ (v− , y) [1−α1(v)]+O(
2
v(log f (y))δ
v ≥ λ+2. (P.2.9)
We claim next that, by (P.2.9)
1 + Δ∗ (v, y)
1
1
)
|Δ(v, y)| ≤ [Δ∗ (v, y) + Δ∗ (v − , y)] + O(
2
2
v(log f (y))δ
(P.2.10)
uniformly for v ≥ λ + 2 and if y is sufficiently large. Indeed, we observe that
1
1
1 ∗
[Δ (v, y) + Δ∗ (v − , y)] − [Δ∗ (v, y) α1(v) + Δ∗ (v − , y) (1 − α1 (v))]
2
2
2
1
1
= ( − α1 (v))(Δ∗ (v, y) − Δ∗ (v − , y))
2
2
and this quantity is positive; for
1
(Δ∗ (v, y) − Δ∗ (v − , y)) ≥ 0
2
by the monotonicity of Δ∗ , and
v
1
1
λ
1
α1 (v) =
jλ (t)dt + α(v) ≤ +
≤ ,
v.jλ (v) v−1
v 2v
2
v ≥ (λ + 2).
This proves (P.2.10).
In order to show that (P.2.4) holds for v ≥ λ + 2, first suppose that (v − 12 ) ≤ v ≤ v.
Let A denote the O-constant in (P.2.10). By the monotonicity of Δ∗ and since
1
v− 12
=
1
1
v 1−(2v)−1
≤ 43 . v1 for v ≥ (λ + 2) > 2
1
1 + Δ∗ (v, y)
1
]
Δ(v , y) ≤ [Δ∗ (v , y) + Δ∗ (v − , y)] + A[
2
2
v(log f (y))δ
1
v
≤
144
M. S. DAOUD and A. HIDRI
4 1 + Δ∗ (v, y)
1
1
≤ [Δ∗ (v, y) + Δ∗ (v − , y)] + A[
].
2
2
3 v(log f (y))δ
Now, if (λ + 1) ≤ v ≤ (v − 12 ), then
1
1
1
Δ(v , y) ≤ Δ∗ (v − , y) ≤ [Δ∗ (v, y) + Δ∗ (v − , y)]
2
2
2
and it follows, by taking the on the right of (P.2.10), that , uniformly for v ≥ λ + 2
1
1
4 1 + Δ∗ (v, y)
∗
Δ (v, y) = sup Δ(y, v ) ≤ [Δ∗ (v, y) + Δ∗ (v − , y)] + A(
).
2
2
3 v(log f (y))δ
λ+1≤v ≤v
After rearranging terms we arrive at the inequality
1
8 1 + Δ∗ (v, y)
)
Δ∗ (v, y) ≤ Δ∗ (v − , y) + A(
2
3 v(log f (y))δ
which we iterate to get
8 (1 + Δ∗ (v, y)) log v
)
Δ∗ (v, y) ≤ Δ∗ (v0 , y) + A(
3
v(log f (y))δ
where
λ + 3/2 ≤ v0 ≤ λ + 2
By (P.2.3), there exists A1 > 0 such that
Δ∗ (v0 , y) ≤
A1
(log f (y))δ
and thus
Δ∗ (v, y) ≤ A∗ [(1 + Δ∗ (v, y))
log v
], v ≥ (λ + 2)
(log f (y))δ
where A∗ = sup( 83 A, A1 ).
Hence
Δ∗ (v, y).[1 − A∗
log v
log v
] ≤ A∗
, v ≥ (λ + 2).
δ
(log f (y))
(log f (y))δ
With the condition 1 ≤ v ≤ exp( C10 (log(f (y))δ )), we have
[1 − A∗
log v
A∗
]
≥
1
−
(log f (y))δ
C0
and therefore, for C0 > A∗ :
Δ∗ (v, y) This proves the theorem
log v
.
(log f (y))δ
(P.2.11)
145
Average of some multiplicative functions
4
Proof of theorem 2
Lemma A:
Let f ∈ ξλ,δ , then
Sf (x, y) ≤ D
xFf (x, y)
; x>1
log ecx
where D is a suitable constant.
Proof:
All we need here to prove the claim is the following weak consequence of (Ω1 ) and
(Ω2 ):
There exists two constant a > 0 and b > 0 such that:
log f (p) ≤ az; z ≥ 1
(A.1)
f (p)≤z
and
log f (pk )
≤ b.
f (pk )
(A.2)
(p,k)≥2
We have the equation
(log cx)Sf (x, y) =
log(
f (n)≤x
cx
)+
f (n)
P (n)≤y
log f (n) = T + M.
f (n)≤x
P (n)≤y
Since, log z ≤ z − 1 for z > 0, we have : 0 ≤ T ≤ cxFf (x, y) − Sf (x, y).
Also,
M =
log f (pk )
f (m)f(pk )≤x;k≥1
(m,p)=1;P (mp)≤y
≤
P (m)≤y
≤
log f (pk )
f (m)f(pk )≤x
f (m)f(p)≤x
log f (p) +
P (mp)≤y,k≥2
log f (p) +
x
f (m)≤x f (p)≤ f (m)
P (m)≤y
≤ axFf (x, y) +
f (pk )≤x;k≥2
f (pk )≤x;k≥2
Sf (
x
, y) log f (pk )
f (pk )
x
x
Ff (
, y) log f (pk )
k
k
f (p )
f (p )
(A.3)
146
M. S. DAOUD and A. HIDRI
by (A.1) and the obvious inequality Sf (x, y) ≤ xFf (x, y) applied in the second sum.
Hence, by (A.2)
M ≤ axFf (x, y) + x
Ff (
1
≤f (pk )≤1
c
x
log f (pk )
,
y)
+x
f (pk )
f (pk )
Ff (
1<f (pk )≤x
x
log f (pk )
,
y)
f (pk )
f (pk )
≤ axFf (x, y) + bxFf (x, y).
Therefore
Sf (x, y) ≤ D
xFf (x, y)
log ecx
with D = c + a + b.
Proposition A:
For f ∈ ξλ,δ we have :
(log cx)Sf (x, y) =
log f (p) + Oλ (
x
f (m)≤x f (p)≤min( f (m) ,f (y))
x(log f (y))λ
) ; v ≥ 1.
(log x)δ
P (m)≤y
Proof:
We star again from the identity (A.3).Suppose f (y) ≤ cx, we have
M =
log f (p) =
f (m)f(p)≤x
(m,p)=1;P (mp)≤y
=
f (m)≤x
x
f (p)≤min( f (m)
,f (y))
(log cx)Sf (x, y) −
f (m)≤x
x
f (p)≤ f (m)
P (m)≤y
p≤y;(m,p)=1
log f (p) −
P (m)≤y
This gives
log f (p)
log f (p).
f (m)f(p)≤x
p|m;P (mp)≤y
f (m)≤x
P (m)≤y
log f (p)
x
f (p)≤min( f (m)
,f (y))
147
Average of some multiplicative functions
≤T +
log f (p) + f (m)f(p)≤x
p|m;P (mp)≤y
f (m)f(pk )≤x;k≥2
P (mp)≤y
We shall show that T ,M and M have order of
log f (pk ) = T + M + M .
x(log f (y))λ
.
(log x)δ
We have by lemma A
M
= = = f (l)f (pk+1 )≤x;k≥1
(p,l)=1;P (pl)≤y
f (pk+1 )≤x
f (l)≤
p≤y;k≥1
f (pk )≤x
f (p)≤f (y);k≥2
xFf (cx, y)[
x
f (pk+1)
P (l)≤y
xFf (cx, y) log f (p)
log f (p)
x
,
y)
log
f
(p)
Sf (
f (pk )
f (pk )≤x
f (p)≤f (y);k≥2
√
f (pk )≤ x
1
1
log
f
(p)
ecx
k
f (p ) log( f (pk ) )
1
1
log f (p) +
k
f (p ) log( fecx
)
(pk )
f (p)≤f (y);k≥2
√
x<f (pk )≤x
f (p)≤f (y);k≥2
by (2.2.3) and (Ω2 ).
The sum M is, for the same reasons, at most of order
x(log f (y))λ
.
(log x)δ
1
log f (p)]
f (pk )
148
M. S. DAOUD and A. HIDRI
Finally, we estimate T . we see that , by Abel summation and lemma A
x
x
x
dt
log( ) d(Sf (t, y)) =
Sf (t, y)
T =
1
1
t
t
c
x
cx
xFf (x, y)
x(log f (y))λ
Ff (t, y)
dt
dt Ff (x, y)
dt 1
1 log et
log et
log x
(log x)δ
c
c
This proves the lemma.
Proposition B:
Let f ∈ ξλ,δ , then for v ∈]0, 1]
Sf (x, y) = Sf (x) =
1 = λC(f )x(log x)λ−1 + O(x(log x)λ−δ−1 ).
f (n)≤x
Proof:
In a first step, we will show that for v ∈]0, 1]
Sf (x, y) = Sf (x) + O(x(log x)λ−δ−1 ).
Indeed
Sf (x, y) = Sf (x) −
(B.1)
1 = Sf (x) − S1 .
f (n)≤x
∃p/n;p>y
When we rearrange the sum S1 following the largest prime divisor of n, we obtain
S1 =
Sf (
f (y)<f (p)≤cx
x
, p)−
f (p)
f (y)<p≤cx f (m)≤ x
f (p)
P (m)=p
1+
f (pk )≤cx;k≥2
Sf (
x
)
=
K+K
+K
.
f (pk )
p>y
For v ∈]0, 1], K = K = 0. On the other hand, by lemma A, proposition 0 and (Ω2 ), K
is at most
f (pk )≤cx
k≥2
This proves (B.1).
x
1
x
) x(log(ecx))λ−δ−1
ecx Ff (
k
k
f (p ) log( f (pk ) )
f (p )
149
Average of some multiplicative functions
Using (B.1) , (Ω1 ) and proposition 0, the equation established in proposition A becomes
[Sf (x)+O(x(log x)λ−δ−1 )] log cx =
[λ
f (m)≤x
x
ecx −δ
x
xFf (x)
+ O(
(log(
) ))]+O(
)
f (m)
f (m)
f (m)
(log x)δ
P (m)≤y
= λxFf (x) + O(x(log x)λ−δ ) + O(x
f (m)≤x
ecx −δ
1
(log(
) ) ; v ∈]0, 1].
f (m)
f (m)
(B.2)
P (m)≤y
Now, let R denote the error term on the right of (B.2), and it suffices to show that
R x(log x)λ−δ .
Indeed, we have
x
f (m)
1/2
hence
x
ex −δ
dt
x
− 1/2 (log(
) )
≥
δ
(log et)
f (m)
f (m)
f (m)
R
f (m)≤x
x
f (m)
1/2
dt
(log et)δ
cx
dt
x
Sf ( , y)
.
t
(log ect)δ
1/2
P (m)≤y
By lemma A and proposition 0 , R is at most of order
cx
cx
Ff ( xt )
dt
dt
λ−1
x(log
x)
x(log x)λ−δ .
Rx
ecx
δ
δ
t
log(
)
(log
ect)
(log
ect)
1/2
1/2
t
Finally, we deduce that
Sf (x) = λC(f )x(log x)λ−1 + O(x(log x)λ−δ−1 ).
Deduction of Theorem 2 from Theorem 1:
The proof of theorem 2 is obtained by combining theorem 1,lemma A and proposition
A.
From proposition A,we obtain
(log cx)Sf (x, y) =
x
f (m)≤ f (y)
f (p)≤f (y)
P (m)≤y
log f (p)
150
M. S. DAOUD and A. HIDRI
x
<f (m)≤x
f (y)
f (p)≤f (y)
+
log f (p) + Oλ (
x(log f (y))λ
) ; v ≥ 1.
(log x)δ
(1.1)
P (m)≤y
We look at each of the two sums on the right side assuming, for the moment, v ≥ 2.
By (Ω1 )
x
f (m)≤ f (y)
f (p)≤f (y)
log f (p) =
{λf (y) + O(f (y)(log f (y))−δ )}
x
f (m)≤ f (y)
P (m)≤y
P (m)≤y
= λf (y)Sf (
x
, y){1 + O((log f (y))−δ )}
f (y)
and using lemma A, we have
x
, y)
Ff ( f (y)
x
, y) x
x(log f (y))λ−1
f (y)Sf (
f (y)
log( fecx
)
(y)
(since Ff (x, y) (log f (y))λ and log f (y) ≤ log( fecx
) for v ≥ 2 ).
(y)
Also, by (Ω1 ), we obtain
x
<f (m)≤x
f (y)
f (p)≤f (y)
log f (p) =
{λ
x
<f (m)≤x
f (y)
P (m)≤y
x
ecx −δ
x
+ O(
(log(
) ))}
f (m)
f (m)
f (m)
P (m)≤y
= λx{Ff (x, y) − Ff (
x
, y)} + O(
f (y)
x
<f (m)≤x
f (y)
ecx −δ
x
(log(
) )).
f (m)
f (m)
P (m)≤y
Now, we will show that the above error term, is at most
f (y)
dt
x
Sf ( , y)
.
t
(log et)δ
1/2
Indeed, we have
x
f (m)
1/2
and for 1 ≤
x
f (m)
ex −δ
x
dt
− 1/2)(log(
) )
≥(
δ
(log et)
f (m)
f (m)
< f (y)
(
x
1/2
x
x
− 1/2) =
(1 − x ) ≥
(1 − 1/2)
f (m)
f (m)
f
(m)
f (m)
(1.2)
151
Average of some multiplicative functions
hence
x
x
(
− 1/2).
f (m)
f (m)
Therefore
x
ex −δ
(log(
) )
f (m)
f (m)
x
f (m)
1/2
dt
(log et)δ
Thus, we get
x
<f (m)≤x
f (y)
ex −δ
x
(log(
) )
f (m)
f (m)
x
<f (m)≤x
f (y)
P (m)≤y
x
f (m)
1/2
dt
(log et)δ
f (y)
1/2
dt
x
Sf ( , y)
.
t
(log et)δ
P (m)≤y
Using lemma A, the integral above it is, in turn, at most of order
f (y)
f (y)
Ff ( xt , y) dt
dt
dt
x
(log f (y))λ f (y)
Sf ( , y)
x
x
ecx
ecx
δ
δ
t
(log et)
t log( t ) (log t)
log( f (y) ) 1/2 t(log et)δ
1/2
1/2
(log f (y))λ
(log f (y))1−δ x(log f (y))λ−δ , v ≥ 2.
x
log f (y)
The main term on the right side of (1.2) is equal to
λA(f )x(log f (y))λ[jλ (v) − jλ−1 (v)] + O(x(log f (y))λ−δ log v)
(by theorem 1).
Thus, from (1.1), we obtain
(log cx)Sf (x, y) = λA(f )x(log f (y))λ[jλ (v) − jλ−1 (v)] + O(x(log f (y))λ−δ log v)
witch reduces, after division by log cx
Sf (x, y) = A(f )x(log f (y))λ−1{ρλ (v) + O(
log v
)}; v ≥ 2.
(log f (y))δ
Now suppose 1 < v < 2. Again (1.1) becomes by (Ω1 ) and proposition B
x(log f (y))λ
x
x
)[1+((log f (y))−δ )]+λx[Ff (x, y)−Ff (
)]+O(
)
f (y)
f (y)
(log x)δ
x
x
(log(
))−δ ); v ∈]1.2].
(1.3)
+O(
f
(m)
f
(m)
x
Sf (x, y) log cx = λf (y)Sf (
f (y)
<f (m)≤x
P (m)≤y
152
M. S. DAOUD and A. HIDRI
We note that the last error term on the right side is at most of the same order as the
error term on the right side of (1.2).
Together with theorem 1 and proposition B, we now can deduce from (1.3) that
Sf (x, y) log cx = λA(f )x(log f (y))λ[jλ (v) − jλ (v − 1)] + O(x(log f (y))λ−δ )) ; v ∈]1.2]
and thus
= A(f )x(log f (y))λvρλ (v) + O(x(log f (y))λ−δ )) ; v ∈]1.2]
(since ρλ (v) = jλ (v) ).
After division by log cx, we get
Sf (x, y) = A(f )x(log f (y))λ−1[ρλ (v) + O(
log v
)] ; v ∈]1.2].
(log f (y))δ
This proves theorem 2.
References
[1] N.G. de Bruijn, On the number of positives integers ≤ x and free of prime factors ¿
y, Nederl. Akad. Wetensch. Proc. Ser . A 54 (1951), 50-60 ; 69 (1966), 239-247.
[2] K.Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark Math. Astr. Fys. (1930), 1-14.
[3] D.Hensley, The convolution powers of the Dickman function, J.London Math. Soc.
(2) 33 (1986) , 395-406.
[4] A.Hildebrand, On the number of positive integers ≤ x and free of prime factor ≥ y,
Journal of Number Theory, 22 (1986), 289-307.
[5] A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Theor.
Nombres Bordeaux 5 (1993), 209-251.
Average of some multiplicative functions
153
[6] M.Naimi, Repartition en moyenne de certaines fonctions arithmetiques sur l’ensemble
des entiers sans grand facteur premier , Journal de Theorie des nombres de Bordeaux
15 (2003) , 745-766
[7] E.Saias, Sur le nombre des entiers sans grand facteur premier, Journal of Number
Theory. 32 (1989) , 78-99 .
[8] H.Smida, Sur les puissances de convolution de la fonction de Dickman, Acta Arith.
59 (1991), 124-143.
[9] J. M. Song, Sums of nonnegative multiplicative functions over integers free of large
prime factors, Acta Arith , 1997, 4 , 353-360.
[10] G. Tenenbaum and J. Wu, Moyenne de certaines fonctions arithmetiques sur les
entiers friables,J.Reine Angew Math 564 (2003), 119-166.
Received: August, 2012