Journal of Inequalities in Pure and
Applied Mathematics
ON THE SYMMETRY OF SQUARE-FREE SUPPORTED
ARITHMETICAL FUNCTIONS IN SHORT INTERVALS
volume 5, issue 2, article 33,
2004.
GIOVANNI COPPOLA
DIIMA-University of Salerno
Via Ponte Don Melillo
84084 Fisciano(SA) - ITALY.
EMail: gcoppola@diima.unisa.it
Received 17 March, 2003;
accepted 02 April, 2004.
Communicated by: A. Fiorenza
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
059-04
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Abstract
We study the links between additive and multiplicative arithmetical functions,
say f, and their square-free supported counterparts, i.e. µ2 f (here µ2 is the
square-free numbers characteristic function), regarding the (upper bound) estimate of their symmetry around x in almost all short intervals [x − h, x + h].
2000 Mathematics Subject Classification: 11N37, 11N36
Key words: Symmetry, Square-free, Short intervals.
The author wishes to thank Professor Saverio Salerno and Professor Alberto Perelli
for friendly and helpful comments. Also, he wants to express his sincere thanks
to Professor Henryk Iwaniec, for his warm and familiar welcome during his stay in
Rutgers University as a Visiting Scholar (the present work was conceived and written
during this period).
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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Contents
Contents
1
Introduction and Statement of the Results . . . . . . . . . . . . . . . . 3
2
Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
Proof of the Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
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1.
Introduction and Statement of the Results
In this paper we study the symmetry, in almost all short intervals, of square-free
supported arithmetical functions.
In our previous paper [3] we applied elementary methods, i.e. the Large
Sieve, in order to study the symmetry of distribution (around x) of the squarefree numbers in "almost all" the "short" intervals [x−h, x+h] (as usual, "almost
all" means for all x ∈ [N, 2N ], except at most o(N ) of them; "short" means that
h = h(N ) and h → ∞, h = o(N ), as N → ∞).
As in [1], [2], [4], and [5] on (respectively) the prime-divisors function, von
Mangoldt function, the divisor function and a wide class of arithmetical functions, we study the symmetry of our arithmetical function f .
def
def
We define the "symmetry sum" of f as (here sgn(t) = t/|t|, sgn(0) = 0)
X
def
Sf± (x) =
f (n)sgn(n − x),
|n−x|≤h
and its mean-square as the "symmetry integral" of f :
2
X X
def
.
If (N, h) =
f
(n)sgn(n
−
x)
x∼N |n−x|≤h
Here and hereafter x ∼ N stands for N < x ≤ 2N .
We will connect (in Theorem 1.1 and Theorem 1.2) If (N, h) and Iµ2 f (N, h),
for suitable f ; thus relating the symmetry of f to that of f on the square-free
numbers (µ2 being their characteristic function). Thus, we can estimate just one
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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symmetry integral for two arithmetical functions, whenever they agree on the
square-free numbers.
As an example, for d(n) the divisor function, [4] estimates Id (N, h); then
(using Theorem 1.4 to check the symmetry of d(n) in arithmetic progressions)
in Theorem 1.3 we bound Iµ2 d (N, h) = Iµ2 2Ω (N, h), and then obtain information on I2Ω (N, h) by Theorem 1.1 (here the function 2Ω(n) is completely
multiplicative, with 2Ω(p) = 2).
We denote with F the set of arithmetical functions f : N → C and with B
the set of f ∈ F, with |f | bounded (by an absolute constant); M denotes the
multiplicative f ∈ F and A the additive ones.
Also, we can define (∀α ∈]1, 2]) the set of "symmetric" arithmetical functions f as (where we assume: ∀E > 0 supN |f | N E ):
N hα
def
cε
Sα = f ∈ F : sup If (N, h, k, q) 2 ε ∀k ≤ N , for some c, ε > 0
k N
q≤N cε
(the -constant is absolute, as well as c > 0), where we have set
2
X
X
x def
If (N, h, k, q) =
f (n)sgn n −
;
k x∼N |n−x/k|≤h/k
n≡0(q)
in the following, as here, we willl abbreviate n ≡ a(q) to mean n ≡ a(mod q).
def
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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We start giving a first link between f and µ2 f (in the sequel L = log N ):
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Theorem 1.1. Let N, h √∈ N, where h = h(N ), h/L2 → ∞ and h = o(N ) as
N → ∞. Assume J Lh , J → ∞ as N → ∞. Let kf k∞ := supN |f |.
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If f is completely multiplicative then
X
2
If (N, h) L max
(i)
DJ
2
d Iµ2 f
d∼D
N h
,
d2 d2
N h
,
d2 d2
+
N h2
kf k2∞
J2
+
N h2
kf k2∞ .
J2
and
2
Iµ2 f (N, h) L max
(ii)
DJ
X
2
d If
d∼D
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
If f is completely additive then
2
(i) If (N, h) L max
DJ
X
2
d Iµ2 f
d∼D
N h
,
d2 d2
+
√ 2
N h2
+ N J hL kf k2∞
2
J
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and
(ii)
Giovanni Coppola
2
Iµ2 f (N, h) L max
DJ
X
d∼D
2
d If
N h
,
d2 d2
+
N h2
2
+ N JL kf k2∞ .
J2
We generalize Theorem 1.1 to additive and to multiplicative functions:
Theorem 1.2. Let f ∈ A ∪ M. Let N, h be natural numbers, with h = N θ
(for 0 < θ < 1). Assume that f is supported over the cube-free numbers and
that ∀E > 0, kf k∞ N E , as N → ∞. Choose ∀α ∈]1, 2] ε = θ(α−1)
> 0.
3
Then
f ∈ Sα ⇔ µ 2 f ∈ Sα .
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We give a concrete example: the function f (n) = 2Ω(n) (where Ω(n) is the
total number of prime divisors of n); in this case f ∈ Sα and µ2 f ∈ Sα ∀α > 23 ,
as we will prove directly, also to detail the (more delicate) estimates
√ Theorem 1.3. Let N, h ∈ N, h = h(N ) ≥ L and h = o LN as N → ∞.
Then
2
X X
Ω(n)
N h3/2 N ε
2
sgn(n
−
x)
x∼N |n−x|≤h
and
2
X X
2
Ω(n)
µ (n)2
sgn(n − x) N h3/2 N ε .
x∼N |n−x|≤h
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
Remark 1.1. We explicitly remark that these bounds are non-optimal.
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This result is obtained directly upon estimating the mean-square of the symmetry sum for the divisor function over the arithmetic progressions:
√ Theorem 1.4. Let N, h ∈ N, with h = h(N ) → ∞ and h = o LN as
N → ∞. Then, uniformly ∀q ∈ N,
2
X X
d(n)sgn(n − x) N hL3 + N L2 log2 q,
|n−x|≤h
x∼N n≡0(q)
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where the O-constant does not depend on q.
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The paper is organized as follows
• In Section 2 we give the necessary lemmas;
• In Section 3 we prove our theorems.
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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2.
Lemmas
def
Lemma 2.1. Let f ∈ F be an arithmetical function, kf k∞ = supN |f (n)|.
Then, for N, h = h(N ) ∈ N and h → ∞, h = o(N ) as N → ∞:
2
X X
X
x
b(m)f (md2 )sgn m − 2 N hL2 kf k2∞ ,
a(d)
√
d
√
x∼N 2h<d≤ x+h
|m−dx2 |≤ dh2
uniformly ∀a, b ∈ B.
(Actually, for our purposes, kf k∞ =
max
N −h≤n≤2N +h
|f (n)|).
Proof. Let Σ be the LHS. By a dyadic dissection and Cauchy inequality
2
X X
X
x 2
2
Σ L √ max√
a(d)
b(m)f (md )sgn m − 2 d hD N
x
h
x∼N d∼D
|m− d2 |≤ d2
2
X X X
x 2
2
L √ max√ D
b(m)f (md )sgn m − 2 d hD N
x∼N d∼D |m− x |≤ h
d2
d2
X
X
X
kf k2∞ L2 √ max√ D
1.
hD N
d∼D
N −h
N <x≤2N
≤m1 ,m2 ≤ 2N 2+h
d2
d
m1 d2 −h≤x≤m1 d2 +h
m2 d2 −h≤x≤m2 d2 +h
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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Clearly, the limitations on x imply m1 − 2h
≤ m2 ≤ m1 + 2h
(here we "reflect"
d2
d2
√
2
the "sporadicity") and this in turn, due to D h ⇒ d h, gives (∀m1
FIXED) O(1) possible values to m2 . Hence Σ is bounded by
X
X
X
kf k2∞ hL2 √ max√ D
1 N hL2 kf k2∞ .
hD N
d∼D
N −h
≤m1 ≤ 2N 2+h
d2
d
|m2 −m1 |1
def
Lemma 2.2. Assume f ∈ F is completely additive and kf k∞ = sup√
N |f |. Let
N, h ∈ N with h = h(N ) → ∞, h = o(N ), as N → ∞. Then ∀J ≤ 2h
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
2
X
X X
x 2
b(m)f (md )sgn m − 2 a(d)
√
d h
x
x∼N d≤ 2h
|m− d2 |≤ d2
2

X
X
X
x 
2
2
L max D kf k∞
b(m)sgn m − 2 DJ
d d∼D x∼N |m− x |≤ h
d2
d2
2 
XX X
x  N h2
+
b(m)f (m)sgn m − 2  + 2 kf k2∞ ,
d J
d∼D x∼N |m− x |≤ h
d2
d2
Giovanni Coppola
uniformly ∀a, b ∈B (bounded arithmetical functions).
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Proof. Let us call the left mean-square Σ. Then Σ is at most
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2
X
X
X
N h2
x
b(m)f (md2 )sgn m − 2 + 2 kf k2∞ .
L2 max a(d)
DJ d J
x∼N
d∼D
|m− dx2 |≤ dh2
Since f is completely additive
2

X
X
X
x

Σ L2 max D 
b(m)f (m)sgn m − 2 DJ
d
x∼N d∼D |m− x |≤ h
d2
d2
2 
X
X
X
x
 N h2
b(m)sgn m − 2  + 2 kf k2∞ ,
+ kf k2∞
d J
x∼N d∼D |m− x |≤ h
2
2
d
d
by the Cauchy inequality. The lemma is thus proved.
Lemma 2.3. Let f be completely multiplicative. Then, if√N, h ∈ N, with h =
h(N ) → ∞ and h = o(N ) (as N → ∞ ), we have ∀J ≤ 2h
2
X X
X
x
a(d)
b(m)f (md2 )sgn m − 2 √
d
x∼N d≤ 2h
|m− dx2 |≤ dh2
2


2
XX X
x Nh 

kf k2∞ L2 maxD
b(m)f (m)sgn m − 2 + 2 
DJ
d J
d∼Dx∼N |m− x |≤ h
2
2
d
d
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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uniformly ∀a, b ∈B.
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Proof. Let us call the left mean-square Σ. Then
2
X
X
X
x
Σ
L2 max a(d)
b(m)f (md2 )sgn m − 2 DJ d x∼N
d∼D
|m− dx2 |≤ dh2
+
N h2
kf k2∞ ,
J2
and being f completely multiplicative we get
2
X
X
X
x
b(m)f (m)sgn m − 2 Σ kf k2∞ L2 max D
DJ
d d∼D x∼N |m− x |≤ h
d2
d2
+
by the Cauchy inequality. The lemma is thus proved.
N h2
kf k2∞ ,
J2
√
Lemma 2.4. Let N, h, J and D be as in Lemma 2.2, with D = o( h). Then
2
XX X
x f (m)sgn m − 2 d d∼D x∼N |m− x |≤ h
d2
d2
2
2
X X X
h
2
+ N D kf k2∞ .
d
f (m)sgn(m − y) +
D
d∼D
y∼ N |m−y|≤h/d2
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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d2
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2
Proof. Write
d2 ) and let Σ be the left mean-square; since
P x = yd
P+ r (0 ≤ r <
2
we have
x∼N =
y∼ N2 +O (d ), then
d
2
X X X X
r h2
f (m)sgn m − y − 2 +
kf k2∞
Σ
d
D
d∼D 0≤r<d2 y∼ N |m−y− r |≤ h
d2
d2
d2
2
(thus hD is due to x-range remainders); then correcting O(1) values of the msum gives as a remainder (due to h-range)
!
X N
2
2
O
d 2 kf k∞ = O N D kf k2∞ .
d
d∼D
Gathering the estimates we then obtain the lemma.
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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3.
Proof of the Theorems
We start by proving Theorem 1.1.
Proof. In both cases (f completely additive or completely multiplicative) we
use the hypothesis on f to "separate variables" after having expressed the symmetry of f by that of µ2 f (for i), say) and the symmetry of µ2 f by that of f (for
ii), say). Thus, to prove i) it will suffice to remember that each natural number
n = md2 , where m and d are natural and µ2 (m) = 1, i.e. m is square-free:
X
X
X
x
f (n)sgn(n − x) =
µ2 (m)f (md2 )sgn m − 2 .
d
√
|n−x|≤h
d≤ x+h |m− x2 |≤ h2
d
d
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
Instead, to prove ii) we simply use the following formula (see [7]):
X
µ2 (n) =
µ(d)
∀n ∈ N
d2 |n
to get
X
|n−x|≤h
µ2 (n)f (n)sgn(n − x) =
X
√
d≤ x+h
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X
µ(d)
|m− dx2 |≤ dh2
x
f (md2 )sgn m − 2 .
d
As for the additional terms in the completely additive case, they come from
the estimate of the square-free symmetry sum as in [3].
Putting together Lemmas 2.1, 2.2, 2.3 and 2.4, the theorem is proved.
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We now come to the proof of Theorem 1.2.
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Proof. We first prove that f ∈ S ⇒ µ2 f ∈ S.
As before, we split at D (to be chosen); say (here [a, b] is the l.c.m. of a, b)
def
Σ=
X
X
µ(d)
|
d≤D
f (n)sgn(n − x)
|
n− x ≤ h
k
k
n≡0([q,d2 ])
=
X
µ(d)
X
X
t|[q,d2 ]
m− x ≤ h
kt2 g kt2 g
g=[q,d2 ]/t (m,g)=1
d≤D
x
f (mt g)sgn m − 2
kt g
2
and observe that, since f is supported over the cube-free numbers, Σ is
X
X
X
X
x
2
µ(d)
f (t g)
µ(j)
f (m)sgn m − 2
kt g
2
d≤D
t|[q,d ]
j|g
g=[q,d2 ]/t
m− x ≤ h
kt2 g kt2 g
m≡0(j)
X1
X
x
δ
kf k∞ N
d max2 f (m)sgn m −
,
2
d j,t≤qd kt[q, d ] d≤D
m− x 2 ≤ h 2
kt[q,d ] kt[q,d ]
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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by (see [7]) the estimate ∀δ > 0 d(n) nδ ; using the hypothesis f ∈ Sα we
get, by Cauchy inequality
Close
X
x∼N
|Σ|2 kf k2∞ N 2δ
α
X 1 X
N hα
2 Nh
d
d2 d≤D k 2 d4 N ε
k2N ε
d≤D
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Hence, it remains to prove that the mean-square of, say
def
X
Σ0 =
√
|
f (n)sgn(n − x)
|
n− x ≤ h
k
k
n≡0([q,d2 ])
D<d≤ x+h
is
X
µ(d)
N hα
|Σ | 2 ε .
k N
x∼N
X
0 2
By the Cauchy inequality and a "sporadicity" argument as in the proof of Lemma 2.1,

X
|Σ0 |2 kf k2∞
x∼N
X

x∼N
X
D<d≤
√h
2
h

+1 
2
kd
Giovanni Coppola
k
2

+ kf k2∞ L2 √ max √ J
h
J
k
δ
N N
Hence
h2
h
+
2
2
k D
k
N hα
|Σ | 2 ε
k N
x∼N
X
0 2
N
XX

d∼J x∼N
+ N δ √ max √ J
h
J
k
N
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
X
m− k[dx2 ,q] ≤ k[dh2 ,q]
Contents

1
X
X
d∼J
N −h
<m≤ 2N2+h
k[d2 ,q]
k[d ,q]
N δ+ε h2−α
1−α δ+ε
+h N k .
D2
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In order to obtain the above required estimate we need ε ≤ θ(α−1)
(for the II
3
0
term in brackets) and, comparing the mean-squares of Σ and of Σ , we come to
4−α
the choice D = N 2(α−1) ε (I term). This proves the first implication.
As for the reverse implication µ2 f ∈ S ⇒ f ∈ S we do not need the
hypothesis on the support of f and we use the same method (but using n = md2
instead of the identity for µ2 ). This finally proves Theorem 1.2.
We now prove Theorem 1.4.
Proof. First of all, let us call Iq (N, h) the mean-square to evaluate.
We will closely follow the proof of Theorem 1 in [4].
In fact, we start from the "flipping" property to write:
X
d(n)sgn(n − x)
|n−x|≤h
n≡0(q)
Giovanni Coppola
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

1X X
h
 X 
=
eq (rn) 2
1 sgn(n − x) + O √ + 1 ,
q r≤q
N
d|n
|n−x|≤h
√
d≤ n
having used the orthogonality of the additive characters (see [7]). By our hypothesis on h (see [4] for the details)
X
2X X X
d(n)sgn(n − x) =
eq (rn)sgn(n − x) + O(1)
q r≤q √ |n−x|≤h
|n−x|≤h
n≡0(q)
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
d≤ x
n≡0(d)
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(here the constant is independent of q, like all the others following).
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Next, write n − x = s to get (again by orthogonality)
X
X
eq (rn)sgn(n − x) = eq (rx)
eq (rs)sgn(s)
|n−x|≤h
n≡0(d)
|s|≤h
s≡−x(d)
X
eq (rx) X
ed (jx)
eq (rs)ed (js)sgn(s)
d j≤d
|s|≤h
X
= eq (rx)
cj,d (q, r)ed (jx),
=
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
j≤d
say, where
def 2i
cj,d (q, r) =
d
X
s≤h
sin 2πs
r j
+
q d
.
Giovanni Coppola
Here (w.r.t. the quoted [4, Theorem 1]) we have the dependence of the Fourier
coefficients on q and r; also, while cd,d = 0 there, here (by the estimate in of [6,
Chap. 25])
2πsr
q
2i X
sin
.
cd,d (q, r) =
d s≤h
q
rd
Hence, this term’s contribute to the mean-square Iq (N, h) is:
2
X 1 X
X
X
eq (rx)
cd,d (q, r)ed (jx) q
√
x∼N r≤q
x∼N
d≤ x
X1
r≤q
r
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!2
L
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2
2
N L log q
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(that is why we have this additional remainder, here!).
J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004
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Henceforth, we can rely upon the proof of [4, Theorem 1], the only difference being the r, s dependence:
(*)
2
X 1 X
XX
c
(q,
r)e
(jx)
e
(rx)
j,d
d
q
q
√
x∼N r≤q
d≤ x j<d
2
X
X
X
X
1
cj,d (q, r)ed (jx)
q r≤q x∼N √ j<d
d≤ x
(we have used the Cauchy inequality).
We apply, then, exactly the same estimates; while there we get (we are quoting inequalities to ease comparison)
X
j<d
|cj,d |2 ≤
X
|cj,d |2 ≤
j≤d
2h
,
d
here we have (the constant c > 0 is ininfluent)
X r j
1 X
|cj,d (q, r)| = c 2
sgn(s1 )sgn(s2 )
e (s1 − s2 )
+
d
q
d
j≤d
j≤d
|s1 |,|s2 |≤h
X
c X
=
sgn(s1 )
sgn(s2 )eq (r(s1 − s2 )),
d
|s |≤h
X
2
|s1 |≤h
2
s2 ≡s1 (d)
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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whence, by (*), we get (see [4, Theorem 1]), ignoring the remainder O(N L2 log2 q):
X 1 X
X
1X
N L2
sgn(s1 )
sgn(s2 )eq (r(s1 − s2 ))
Iq (N, h) q r≤q
d
√
|s |≤h
d≤ 2N
= N L2
|s1 |≤h
2
s2 ≡s1 (d)
X
X 1 X
sgn(s1 )
sgn(s2 )
d
√
|s |≤h
d≤ 2N
|s1 |≤h
2
s2 ≡s1 (d)
s2 ≡s1 (q)


 X 1

N L2 
h+
 h d
d≤
L
[d,q]≤ h
L
X
h
<d≤
L
√
2N
1
d
2

h

+ h .
d

On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
Thus
Iq (N, h) N hL3 + N L2 log2 q.
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Contents
We now prove Theorem 1.3.
Proof. We first show the second estimate.
First of all, we observe that µ2 (n)2Ω(n) = µ2 (n)d(n), ∀n ∈ N; here we will
apply the flipping property of the divisor function as in [4].
Then, we will try to link our symmetry integral (for µ2 2Ω ) with that of d(n).
Writing µ2 (n) as before
X
X
X
µ2 (n)d(n)sgn(n − x) =
µ(d)
d(n)sgn(n − x).
|n−x|≤h
√
d≤ x+h
|n−x|≤h
n≡0(d2 )
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Splitting the range at D = D(x) ≤
def
Σ1 (x) =
X
√
µ(d)
x + h (to be chosen later), we treat, say
X
d(n)sgn(n − x)
|n−x|≤h
n≡0(d2 )
d≤D
by the Cauchy inequality and Theorem 1.4 to get
2
X
X
X
X
|Σ1 (x)|2 D
d(n)sgn(n − x)
|n−x|≤h
x∼N
d≤D x∼N 2
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
n≡0(d )
2
3
N D L (h + L) N D2 hL3 ,
Giovanni Coppola
by our hypothesis on h. It remains to bound the mean-square of, say
Title Page
def
X
Σ2 (x) =
√
D<d≤ x+h
µ(d)
X
d(n)sgn(n − x).
Contents
|n−x|≤h
n≡0(d2 )
√
We split again at 2h (to distinguish non-sporadic and sporadic terms).
Since by the classical estimate d(n) nε (see [7]; here ε > 0 will not be
the same at each occurrence) we estimate trivially (the non-sporadic terms)
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X
√
D<d≤ 2h
µ(d)
X
|n−x|≤h
n≡0(d2 )
d(n)sgn(n − x) X
√
D<d≤ 2h
ε
2
hN
Nh ε
N
2
d
D2
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J. Ineq. Pure and Appl. Math. 5(2) Art. 33, 2004
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we get, together with (the sporadic terms, treated by Lemma 2.1)
2
X
X X
µ(d)
d(n)sgn(n
−
x)
N hN ε ,
√
√
|n−x|≤h
x∼N 2h<d≤ x+h
2
n≡0(d )
that
X
2
|Σ2 (x)| x∼N
N h2
+ N h N ε.
D2
Thus, comparing the mean-squares of Σ1 (x) and Σ2 (x) we make the best choice
D = h1/4 , finally proving the second estimate.
Writing I2Ω for the symmetry integral of 2Ω , we apply Theorem 1.1 to this
function; then, i) gives us
N h3/2
N h2
I2Ω (N, h) L2 max
d2 2 3 N ε + 2 N ε N h3/2 N ε ,
DJ
d d
J
d∼D
X
by the choice J =
Theorem 1.3.
√
h. This gives the first estimate, hence finally proving
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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References
[1] G. COPPOLA, On the symmetry of distribution of the prime-divisors function in almost all short intervals, to appear.
[2] G. COPPOLA, On the symmetry of primes in almost all short intervals,
Ricerche Mat., 52(1) (2003), 21–29.
[3] G. COPPOLA, On the symmetry of the square-free numbers in almost all
short intervals, submitted.
[4] G. COPPOLA AND S. SALERNO, On the symmetry of the divisor function
in almost all short intervals, to appear in Acta Arithmetica.
[5] G. COPPOLA AND S. SALERNO, On the symmetry of arithmetical functions in almost all short intervals, submitted.
[6] H. DAVENPORT, Multiplicative Number Theory, Springer Verlag, New
York 1980.
[7] G. TENENBAUM, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 46, Cambridge University Press, 1995.
On the Symmetry of
Square-Free Supported
Arithmetical Functions in Short
Intervals
Giovanni Coppola
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