On the average of some arithmetical functions under a ... - Project Euclid

Hiroshima Math. J. 46 (2016), 37–54

On the average of some arithmetical functions under a constraint on the sum of digits of squares Dedicated to my parents He´di and Wiem for their endless support, with love Karam Aloui (Received January 6, 2014) (Revised June 5, 2015)

Abstract. Let q b 2 be an integer and Sq ðnÞ denote the sum of the digits in base q of the positive integer n. We look for an estimate of the average of some multiplicative arithmetical functions defined by sums over divisors d of n satisfying Sq ðd 2 Þ 1 r mod m for some integers r and m.

1. Introduction Throughout this paper, we denote by N, N0 , Z, R and C the sets of positive integers, non negative integers, integers, real and complex numbers respectively. Given a real number x, bxc denotes the greatest integer a x and eðxÞ ¼ e 2ipx . The greatest common divisor of two integers a and b will be denoted by ða; bÞ and if a a b we denote by wa; bx the set fa; a þ 1; . . . ; bg. The number of distinct prime factors of a positive integer n will be denoted oðnÞ. First, we shall introduce the following definition: let n A N0 and q be an integer b 2. The sequence ðaj ðnÞÞj A N0 A f0; 1; . . . ; q
1gN0 is defined to be the unique sequence satisfying n¼

y X

ak ðnÞq k :

ð1:1Þ

k¼0

The right hand side of the expression (1.1) shall be called the expansion of n to the base q. We shall set SðnÞ ¼ Sq ðnÞ ¼

y X

ak ðnÞ:

k¼0

2010 Mathematics Subject Classification. 11N37, 11N60, 11N69. Key words and phrases. sum of digits function, multiplicative functions.

38

Karam Aloui

A function f : N0 ! C is called completely q-additive if f ð0Þ ¼ 0 and f ðaq k þ bÞ ¼ f ðaÞ þ f ðbÞ for any integers a b 1, k b 1 and 0 a b < q k . Such functions were introduced by Gelfond [7] and further studied by Delange [5], Be´sineau [3], Coquet [4], Ka`tai [9] and others. Using the base q expansion (1.1), we find that the function f is completely q-additive if and only if f ð0Þ ¼ 0 and f ðnÞ ¼

y X

f ðak ðnÞÞ:

k¼0

It follows that a completely q-additive function is completely determined by its values on the set f0; 1; . . . ; q
1g. A typical example of a completely q-additive function is the sum of digits function S. Another kind of arithmetic functions is the multiplicative ones, i.e. that satisfy f ð1Þ ¼ 1 and whenever a and b are coprime integers, then f ðabÞ ¼ f ðaÞ f ðbÞ (see [2, chapter 2] for further informations). In this paper, we shall focus on the following functions depending on a positive integer n: P  the number of positive divisors function, t : n 7! 1. djn



The sum of the s-th powers of all the positive divisors function (for
 P n s . In particular, s0 ¼ t. s A R), ss : n 7! djn d P  The number of positive integers a n and coprime to n, j : n 7! 1. 1akan  The Mo ¨ bius function, ðk; nÞ¼1 8 > < 1; if n ¼ 1; m : n 7! ð
1Þ r ; if n ¼ p1 . . . pr ; a product of distinct primes; > : 0; otherwise: 

The non principal Dirichlet character modulo 4, ( 0; if 2jn; w : n 7! ð
1Þ ðn
1Þ=2 ; otherwise:

 The number of representations of n as the sum of two integral squares denoted by rðnÞ. Except the last one, all these functions are multiplicative and it can be shown (see [8, chapter 16] for instance) that X n jðnÞ ¼ mðdÞ ; d djn

rðnÞ ¼ 4

X djn

wðdÞ:

Arithmetical functions under digital constraint

39

rðnÞ is multiplicative and thus rðnÞ is ‘‘almost In fact, this implies that 4 multiplicative’’. For every r A Z, q and m b 2 such that ðm; q
1Þ ¼ 1, we define the following functions that depend on n, q, r and m (but we will omit the latter ones for briefness) X

tðnÞ ¼

1;

djn Sðd 2 Þ1r mod m

ss ðnÞ ¼

X djn Sðd 2 Þ1r mod m


s n ; d

X

jðnÞ ¼

djn Sðd 2 Þ1r mod m

rðnÞ ¼ 4

for s A R;

n mðdÞ ; d

X

wðdÞ:

djn Sðd 2 Þ1r mod m j

Note that the assumption ðm; q
1Þ ¼ 1 is crucial since it implies that ðq
1Þ m A RnZ for all j A w1; m
1x, a result that allows to use the Theoerem A which will be stated later. P P P Our target, in this paper, is to estimate tðnÞ, ss ðnÞ, jðnÞ and nax nax nax P rðnÞ (representing the averages of the functions t, ss , j and r respectively, nax

in accordance to the study made in [8, chapter 18]). In order to detect the congruences, we shall use the classic orthogonality relation   m
1
1; 1X jða
bÞ e ¼ m j¼0 m 0;

if a 1 b mod m; else:

ðm A N; a; b A ZÞ

ð1:2Þ

Finally, Gelfond [7] alluded the problem of giving an estimate for the number of values of a polynomial P (P takes only integer values on the set N) satisfying the condition SðPðnÞÞ 1 r mod m. Basically, we shall need the following result proved by Mauduit and Rivat [10], answering the question of Gelfond in the case PðnÞ ¼ n 2 . Theorem A [Mauduit-Rivat, 2007]. Let q b 2 be an integer and a A R such that ðq
1Þa A RnZ then there exists sq ðaÞ > 0 and x0 :¼ x0 ðq; aÞ b 2

40

Karam Aloui

such that for every real number x b x0 , we have  
  X log x ð1=2ÞoðqÞþ4 1
sq ðaÞ  5=2 1=2 2  7=2 eðaSðn ÞÞ a 4q ðlog qÞ tðqÞ 1þ x :   nax log q In particular, this means that for x su‰ciently large X

eðaSðn 2 ÞÞ f ðlog xÞ ð1=2ÞoðqÞþ4 x 1
sq ðaÞ :

nax

Recently, Mkaouar and Wanne`s [11] used an improved result of Drmota, Mauduit and Rivat [6], in addition to the classic Abel’s summation formula, to prove interesting results about the average of some additive functions (namely, the number of distinct prime factors o and the total number of prime factors W of a positive integer n) under digital constraints. Using the same ideas, we will follow a similar path, in this article, in order to study the multiplicative functions under constraints on the sum of the digits of squares. Indeed, this is our second approach concerning this topic after the first study done in [1]. 2. A lemma on power sums We need a classical lemma estimating some expressions that will be needed later. In part a), the constant g is Euler-Mascheroni’s constant defined by the equation ! X1 g ¼ lim
log x : x!þy n nax In part b), zðsÞ denotes the Riemann zeta function defined by the equations 8 þy > X > 1 > > ; if s > 1; > s < n n¼1 zðsÞ ¼ ! > X1 > x 1
s > > lim
; if 0 < s < 1; > : x!þy ns 1
s nax a proof can be found in [2, chapter 3]. Lemma 2.1. As x ! þy, we have
 P1 1 a) ¼ log x þ g þ O . x nax n P 1 x 1
s þ zðsÞ þ Oðx
s Þ, if s > 0, s 0 1. b) ¼ s 1
s nax n

Arithmetical functions under digital constraint

41

P 1 ¼ Oðx 1
s Þ, if s > 1. s n>x n P s x sþ1 þ Oðx s Þ, if s b 0. d) n ¼ sþ1 nax

c)

3. Average of t Theorem 3.1. Let q b 2 and m b 2 be integers such that ðq
1; mÞ ¼ 1, let r A Z. Then we have X

tðnÞ ¼

nax

1 a þ 2g
1 x log x þ x þ Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m =2 Þ; m m

x ! þy, where g stands for Euler-Mascheroni’s constant,
 j min sq (sq being the constant stated in Theorem A) and m j A w1; m
1x ! !  ð þy X
m
1
X rj j du 2 Sðd Þ a¼ e
e : m m u2 1 j¼1 dau as

Proof.

Given x large enough, we have X

X

tðnÞ ¼

nax

1

d; l dlax 2 Sðd Þ1r mod m

¼ A1 þ A2
A3 ; where A1 ¼

A2 ¼

X

X

pffiffi lax=d da x 2 Sðd Þ1r mod m

X

X pffiffi la x

1;

1

dax=l Sðd 2 Þ1r mod m

and A3 ¼

X

X

pffiffi pffiffi da x la x Sðd 2 Þ1r mod m

1:

sq; m ¼

42

Karam Aloui

First, we may write using the orthogonality relation (1.2)   X x A1 ¼ pffiffi d da x Sðd 2 Þ1r mod m

¼

   m
1
1 X x X j ðSðd 2 Þ
rÞ e m dapffiffix d j¼0 m

¼

 m
1
pffiffiffi 1 X 1 1 X rj x þ x e
Sj ðxÞ þ Oð xÞ; pffiffi d m m j¼1 m

ð3:1Þ

da x

where X e Sj ðxÞ ¼ pffiffi



da x

j 2 m Sðd Þ

d

;

for each j A w1; m
1x:

The first sum can be estimated as a consequence of Lemma 2.1 giving pffiffiffi 1 X 1 1 g x ¼ x log x þ x þ Oð xÞ: pffiffi d m 2m m

ð3:2Þ

da x

Next, using Abel’s summation formula, we get  X e j Sðd 2 Þ m Sj ðxÞ ¼ pffiffi d da x

pffiffi x


 ð 1 X j Sðd 2 Þ þ ¼ pffiffiffi e x dapffiffix m 1

! X
j du 2 Sðd Þ e : m u2 dau

Thanks to Theorem A, we write for j A w1; m
1x   X
j ðlog uÞ ð1=2ÞoðqÞþ4 1  2 Sðd Þ  2 f e :  u dau m u 1þsq ð j=mÞ Hence, we obtain ! ! ð pxffiffi X
j du ðlog xÞ ð1=2ÞoðqÞþ4 2 Sðd Þ e ¼ aj þ O ; m u2 xsq ð j=mÞ=2 1 dau where ! ð þy X
j du 2 Sðd Þ aj ¼ e ; m u2 1 dau

for every j A w1; m
1x:

ð3:3Þ

Arithmetical functions under digital constraint

Using Theorem A again, we have !
 1 X j ðlog xÞ ð1=2ÞoðqÞþ4 2 pffiffiffi Sðd Þ ¼ O e : x dapffiffix m xsq ð j=mÞ=2

43

ð3:4Þ

Taking the identities (3.3) and (3.4) jointly and setting
 j sq; m ¼ min sq , m j A w1; m
1x we find

! ðlog xÞ ð1=2ÞoðqÞþ4 Sj ðxÞ ¼ aj þ O : xsq; m =2

Considering (3.2) and choosing sq; m small enough, we go back to (3.1) in order to find 1 aþg x log x þ x þ Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m =2 Þ; 2m m
 m
1 P rj where a ¼ e
aj . m j¼1 Next, we write  m
1
1 X X X j ðSðd 2 Þ
rÞ e A2 ¼ m pffiffi dax=l j¼0 m A1 ¼

ð3:5Þ

la x

¼


  m
1
1 X X 1X jr X X j Sðd 2 Þ : 1þ e
e pffiffi m pffiffi dax=l m j¼1 m m dax=l la x

la x

It is easy to check, using Lemma 2.1, that
 1 X X 1 X x þ Oð1Þ 1¼ m pffiffi dax=l m pffiffi l la x

la x

¼

pffiffiffi 1 g x log x þ x þ Oð xÞ: 2m m

ð3:6Þ

As a consequence of Theorem A, we get for j A w1; m
1x, 
1
sq ð j=mÞ X X
j X x Sðd 2 Þ f e ðlog xÞ ð1=2ÞoðqÞþ4 pffiffi pffiffi m l dax=l la x

la x

f ðlog xÞ ð1=2ÞoðqÞþ4 x 1
sq ð j=mÞ=2 : The last bound follows from Lemma 2.1.

ð3:7Þ

44

Karam Aloui

Combining (3.6) and (3.7) and choosing sq; m small enough gives A2 ¼

1 g x log x þ x þ Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m =2 Þ; 2m m

ð3:8Þ

Finally, thanks to Theorem A, we obtain  m
1 X
pffiffiffi 1 X pffiffiffi 1X j 2 ðSðd Þ
rÞ ð x þ Oð1ÞÞ A3 ¼ ð x þ Oð1ÞÞ þ e m pffiffi m j¼1 pffiffi m da x

¼

da x

1 x þ Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m =2 Þ; m

ð3:9Þ

Gathering (3.5), (3.8) and (3.9) together, we get the desired conclusion.

9

4. Average of ss The case s ¼ 0 was considered in the previous paragraph. consider s > 0 and deal carefully with the subcase s ¼ 1.

We shall first

Theorem 4.1. Let q b 2 and m b 2 be integers such that ðm; q
1Þ ¼ 1, let r A Z and s > 0 be a real number. We assert that  zðsþ1Þ X þ b sþ1 Oðx log xÞ; if s ¼ 1; x ss ðnÞ ¼ sþ1 þ as x ! þy; m Oðx t Þ; if s 0 1; nax where z stands for the Riemann zeta function defined in !Lemma 2.1,
!
 m
1 Ð þy P P jr j du Sðd 2 Þ t ¼ maxð1; sÞ and b ¼ e
e . 1 m m u sþ2 j¼1 dau Proof.

Let x be large enough, we may write
s X X X n ss ðnÞ ¼ d nax nax djn Sðd 2 Þ1r mod m

¼

X

X
n s

nax dax Sðd 2 Þ1r mod m n¼hd

¼

X

X

d hs

dax hax=d Sðd 2 Þ1r mod m

¼

X dax Sðd 2 Þ1r mod m

( sþ1 x


s ) x þO s : sþ1 d d

ð4:1Þ

Arithmetical functions under digital constraint

45

Here, we should treat the subcases s ¼ 1 and s 0 1 separately, using in both cases Lemma 2.1 and applying (1.2).  If s ¼ 1, then ! X X X1 1 2 1 s1 ðnÞ ¼ x þO x 2 d2 d nax dax dax Sðd 2 Þ1r mod m

j  m
1
1 2X jr X e m Sðd 2 Þ x ¼ e
þ Oðx log xÞ d2 2m m dax j¼0 ¼





 1 2 1 x
þ zð2Þ þ O x
2 2m x j  m
1
1 2X jr X e m Sðd 2 Þ x þ e
þ Oðx log xÞ: 2m m dax d2 j¼1

If s 0 1, we set t ¼ maxð1; sÞ so that X

1 x sþ1 ss ðnÞ ¼ s þ 1 nax

X dax Sðd 2 Þ1r mod m

X 1 þO x sþ1 d ds dax 1

!

s

j  m
1
X 1 jr X e m Sðd 2 Þ sþ1 x ¼ e
ðs þ 1Þm m dax d sþ1 j¼0
 1
s

 s x
s þ zðsÞ þ Oðx Þ þO x 1
s 

1 x
s sþ1
s
1 x ¼ þ zðs þ 1Þ þ Oðx
Þ ðs þ 1Þm s j  m
1
X 1 jr X e m Sðd 2 Þ sþ1 þ x e
þ Oðx t Þ: ðs þ 1Þm m dax d sþ1 j¼1 Going back to (4.1), we can summarize by the following formula  X Oðx log xÞ; if s ¼ 1; zðs þ 1Þ sþ1 x ss ðnÞ ¼ þBþ ðs þ 1Þm Oðx t Þ; if s 0 1; nax

ð4:2Þ

where j  m
1
X 1 jr X e m Sðd 2 Þ sþ1 x e
: B¼ d sþ1 ðs þ 1Þm m dax j¼1

ð4:3Þ

46

Karam Aloui

We apply Abel’s formula to B in order to get, for j A w1; m
1x, Tj ðxÞ ¼

Xe dax



j 2 m Sðd Þ d sþ1




 ! ðx X
1 X j j du 2 2 Sðd Þ þ ðs þ 1Þ Sðd Þ ¼ sþ1 e e : sþ2 x dax m m u 1 dau As a consequence of Theorem A, the latter integral is again absolutely convergent. Therefore, we get ! ðx X
j du 2 Sðd Þ e ¼ b j þ Oððlog xÞ ð1=2ÞoðqÞþ4 x
s
sq ð j=mÞ Þ; m u sþ2 1 dau

ð4:4Þ

with ! ð þy X
j du 2 Sðd Þ bj ¼ e ; sþ2 m u 1 dau

for every j A w1; m
1x:

Using Theorem A once again, we have
 1 X j 2 Sðd e Þ ¼ Oððlog xÞ ð1=2ÞoðqÞþ4 x
s
sq ð j=mÞ Þ: x sþ1 dax m

ð4:5Þ

Considering the identities (4.4) and (4.5) jointly, we find Tj ðxÞ ¼ ðs þ 1Þbj þ Oððlog xÞ ð1=2ÞoðqÞþ4 x
s
sq ð j=mÞ Þ: Going back to (4.3), we write b sþ1 x þ Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m Þ; m

 m
1 P j jr with sq; m ¼ min sq e
and b ¼ b. m m j j A w1; m
1x j¼1 Hence, taking (4.2) and (4.6) jointly enables to reach our result. B¼

ð4:6Þ

9

In order to find the average order of ss ðnÞ for negative s, we shall set u ¼
s where u > 0. Theorem 4.2. Let q b 2 and m b 2 be integers such that ðm; q
1Þ ¼ 1, let r A Z and s < 0 be a real number. Thus

Arithmetical functions under digital constraint

47

8 > > > Oð1Þ; if v < 0; > < X zð1
sÞ Oððlog xÞ ð1=2ÞoðqÞþ4 x v Þ; if v b 0 and s 0
sq; m ; xþ ss ðnÞ ¼ > Oððlog xÞ ð1=2ÞoðqÞþ5 x 1þs Þ; if sq  j ¼ sq; m ; m > nax > m > : Ej A w1; m
1x and s ¼
sq; m b
1;
 j as x ! þy, where z is the Riemann zeta function, sq; m ¼ min sq and m j A w1; m
1x v ¼ maxð1 þ s; 1
sq; m Þ. Proof. In fact, if we set u ¼
s > 0, it follows from the orthogonality relation (1.2) that X

ss ¼

nax

¼

X

X

nax

djn Sðd 2 Þ1r mod m

X 1 hu hax


u d n

X

1

dax=h Sðd 2 Þ1r mod m

¼ G 1 þ G 2;

ð4:7Þ

where   1X 1 x G1 ¼ m hax h u h and G2 ¼


  m
1
1X rj X 1 X j 2 Sðd e
e Þ : m j¼1 m hax h u dax=h m

Therefore, X 1 1 X 1 G1 ¼ x þO 1þu m hax h hu hax

!

1 ¼ xfzð1 þ uÞ þ Oðx
u Þg þ m 8 > < Oðlog xÞ; zð1 þ uÞ ¼ x þ Oð1Þ; > m : Oðx 1
u Þ;

(

Oðlog xÞ; if u ¼ 1; n 1
u o O x1
u þ zðuÞ þ Oðx
u Þ ;

if u ¼ 1; if u > 1; if u < 1:

else;

48

Karam Aloui

Besides, Theorem A enables us to get for j A w1; m
1x,  
 X 1 X
j X 1
x ð1=2ÞoðqÞþ4 x 1
sq ð j=mÞ 2 Sðd Þ f e log h u dax=h m hu h h hax hax f ðlog xÞ ð1=2ÞoðqÞþ4 x 1
sq ð j=mÞ Thanks to Lemma 2.1, 8 > Oððlog xÞ ð1=2ÞoðqÞþ5 x 1
u Þ; > > > < G2 ¼ > Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
u Þ; > > > : Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m Þ;

X

1

hax

h 1þu
sq ð j=mÞ

:

 if sq mj ¼ sq; m ; Ej A w1; m
1x and u ¼ sq; m ; if u < sq; m ; if u > sq; m :

A discussion on the possible orders of the real numbers 0, 1
u and 1
sq; m completes the proof. 9 5. Average of j Theorem 5.1. Let q b 2 and m b 2 be integers such that ðm; q
1Þ ¼ 1, let r A Z. We assert that X

jðnÞ ¼

3 p2

nax

where r ¼

m
1 P j¼1

þr 2 x þ Oðx log xÞ; m


 Ð þy jr e
1 m




P

j Sðd 2 Þ mðdÞe m dau

as x ! þy; !

! du . u3

Let x be large enough, due to (1.2) and Lemma 2.1 we have X X n jðnÞ ¼ mðdÞ d nax nax djn

Proof. X

Sðd 2 Þ1r mod m

¼

X

mðdÞ

dax Sðd 2 Þ1r mod m

¼

X dax Sðd 2 Þ1r mod m

X

h

hax=d


mðdÞ


 x2 x þ O d 2d 2

  m
1
e mj Sðd 2 Þ 1 2X jr X x ¼ e
mðdÞ þ Oðx log xÞ d2 2m m dax j¼0

Arithmetical functions under digital constraint þy X 1 1 2X mðdÞ 2 x ¼ þ O x 2m d¼1 d 2 d2 d>x

49

!

  m
1
e mj Sðd 2 Þ 1 2X jr X x þ e
mðdÞ þ Oðx log xÞ 2m m dax d2 j¼1 ¼

þy 1 2X mðdÞ x þ C þ Oðx log xÞ; 2m d¼1 d 2

ð5:1Þ

where   m
1
e mj Sðd 2 Þ 1 2X jr X x e
mðdÞ : C¼ 2m m dax d2 j¼1

ð5:2Þ

First, using the Mo¨bius inversion formula (see [2, p. 32]), we get 0 1 0 1 ! ! þy þy þy þy X X X X X X 1 mðmÞ 1 B 1 @ C mðdÞA ¼ mðmÞA ¼ 1; ¼ 2 2 2@ 2 d m k k m¼1 d¼1 k¼1 k¼1 d; m mjk dm¼k

so that þy X mðdÞ d¼1

d2

¼

6 : p2

ð5:3Þ

Next, we may apply Abel’s summation formula again to C in order to obtain Uj ðxÞ ¼

X

mðdÞ

dax

e



j 2 m Sðd Þ 2 d





! ðx X 1 X j j du 2 2 Sðd Þ þ 2 Sðd Þ ¼ 2 mðdÞe mðdÞe : x dax m m u3 1 dau The integral is trivially convergent. ðx 1

Subsequently, we get
! X  j du 2 Sðd Þ mðdÞe ¼ rj þ O x
1 ; 3 m u dau

with rj ¼

ð þy X 1




j Sðd 2 Þ mðdÞe m dau

!

du ; u3

for each j A w1; m
1x:

ð5:4Þ

50

Karam Aloui

Clearly, we have
  1 X j 2 Sðd Þ ¼ O x
1 : mðdÞe x 2 dax m

ð5:5Þ

Considering the identities (5.4) and (5.5) jointly, we find  Uj ðxÞ ¼ 2rj þ O x
1 : Going back to (5.2), we write r 2 x þ OðxÞ; ð5:6Þ m

 m
1 P j jr with sq; m ¼ min sq e
and r ¼ r. m m j j A w1; m
1x j¼1 Hence, considering (5.1), (5.3) and (5.6) together gives the desired result. 9 C¼

6. Average of r Theorem 6.1. Let q b 2 and m b 2 be integers such that ðm; q
1Þ ¼ 1, let r A Z. We state that X

rðnÞ ¼

nax

p þ 4n x þ Oððlog xÞ ð1=2ÞoðqÞþ4 x 1
sq; m =2 Þ; m

as x ! þy;

where
! !  ð þy X m
1
X jr j du 2 n¼ Sðd Þ e
wðdÞe m m u2 1 j¼1 dau and sq; m


 j ¼ min sq (sq being the constant stated in Theorem A). m j A w1; m
1x

Proof. Given x large enough, we have X X X rðnÞ ¼ 4 wðdÞ nax

nax

¼4

djn Sðd 2 Þ1r mod m

X

wðdÞ

dhax Sðd 2 Þ1r mod m

¼4

X dax Sðd 2 Þ1r mod m

wðdÞ

X hax=d

1

Arithmetical functions under digital constraint

X

¼4

wðdÞ

  x d

wðdÞ

  X x þ4 pffiffi d

dax Sðd 2 Þ1r mod m

¼4

X pffiffi da x 2 Sðd Þ1r mod m

ha x

X pffiffi x