The complex sum of digits function and primes

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

ÖRG

M. T

HUSWALDNER

The complex sum of digits function and primes

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{1 (2000),}

p. 133-146

<http://www.numdam.org/item?id=JTNB_2000__12_1_133_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The

_Complex

Sum of

_Digits

Function

and Primes

par

JÖRG

M. THUSWALDNER

RÉSUMÉ. La notion de

_{développement q-adique}

d’un _{entier, pour}

une base _q

_donnée,

se

_généralise

dans l’anneau des entiers de Gauss

_{Z [i]}

au

_{développement}

d’un entier de Gauss suivant une

certaine base b ~

_{Z [i],}

ce

_{développement}

étant _unique. Dans

cet

_article,

on s’intéresse à la fonction 03BDb,

désignant

la somme

de chiffres dans le

_{développement}

suivant la base b. On montre

un résultat sur la fonction somme de chiffres pour les nombres

non

_multiples

d’une _puissance

_f-ième

d’un nombre _premier. On

établit aussi _{pour 03BDb} un théorème du _type Erdös-Kac. Dans ces

résultats, l’équidistribution

de 03BDb

joue

un rôle essentiel. Partant

de

_cela,

les démonstrations font alors

_appel

à des méthodes de

crible,

ainsi

_qu’à

une version du modèle de Kubilius.

ABSTRACT. Canonical number _systems in the

_ring

of Gaussian

integers

Z

_[i]

are the natural

_{generalization}

of

_{ordinary q-adic}

num-ber _systems to

_{Z [i].}

It turns out, that each Gaussian

_integer

has a

unique representation with _respect to the powers of a certain base

number b. In this paper we

_investigate

the sum of

digits

function

03BDb of such number systems. First we _prove a theorem on the sum

of

_digits

of

_numbers,

that are not divisible

_{by the}

_f-th

power of a

prime.

Furthermore,

we establish an Erdös-Kac _type theorem for

03BDb . In all

proofs

the

_{equidistribution of 03BDb}

in residue classes

_plays

a crucial rôle.

Starting

from this fact we use sieve methods and a

version of the model of Kubilius to prove our results.

1. INTRODUCTION

Let

_vq(n)

denote the sum of

_digits

of the

_q-adic

_{representation}

of a

pos-itive

_integer

n. Gelfond

[5]

proved,

that

vq(n)

is

equidistributed

in residue

classes modulo an

_integer.

In

_particular,

he established the

_following

result.

For _{r, m} E Z with

one has

The

_exponent

_~1

1 in the error term can be

_{computed explicitly}

and

does not

_depend

on _{a, s,} r and N. This result forms the basis for the

application

of sieve

_methods,

in order to

_get

results on the sum of

_digits

of

_prime

numbers. The first result of this

_type,

also contained in Gelfond’s

paper

[5],

is the

following.

Let

Sr,m(N) = In

N r

(~n)}.

Then

the number of elements of

_8r,m(N),

_being

not divisible

_by

the

_f-th

_power of a

_prime

is

_{given by}

with a constant

_A2

1.

_{Here (}

denotes the Riemann zeta function.

Re-cently,

Mauduit and

_S6xk6zy

_[15]

_proved

an Erd6s-Kac

_type

theorem for

the elements of the set

_8r,m(N):

Let _m, r E Z such that

₍₁₎

holds. Denote the normal law

_by

_~(a?)

and let

_w(n)

count the distinct

_prime

divisors of

n. Then

The aim of this _paper is to extend these results to the sum of

_digits

function of canonical number

_systems

in the

_ring

of Gaussian

_integers

First we

give

a definition of these number

_systems.

Let b E

and fii

=

{O, 1,... ,

N(b) -

11,

where

_N(b)

denotes the norm of b over

Q.

If _any

number 1

E admits a

representation

of the form

for c; E

_{{0,1, ... ,}

_{N(b) -}

_{11 (0 j}

_h)

and 0 for

_{h ~}

0,

then

_{(b, N)}

is called a canonical number

_system

in

7G~i~.

b is called the base of this

number

_system.

In

_Kitai,

Szab6

_[11]

it is shown that b can serve as a base

of a canonical number

_system

in if and

only

if

The sum of

_digits

function of the number

_system

(b,

N)

is defined

by

Some

_properties

of this function have been

_investigated

in recent _papers.

An

_asymptotic

formula of the

_summatory

function of in

_large

circles

Gittenberger

and Thuswaldner

_[6]

established

_asymptotic

formulas for the

moments of

The notion of canonical number

_systems

can be extended to

_arbitrary

number

_{fields, provided}

that

_they

have a _power

integral

basis

(cf.

Kovics

[12]).

The bases of these number

_systems

are characterized in

Kovics,

Peth6

_[13].

This characterization is not

_explicit

and

_depends

on the

shape

of the

_integral

basis of the number field. Some results on the sum of

digits

function can be extended to the

_general

case of canonical number

_systems

in number fields. The results

₍₃₎

and

₍₄₎

can be

_generalized

to number

fields,

whose

_ring

of

_integers

is a

_unique

factorization domain. For further

generalizations

a new notion of the sum of

digits

function with ideals as

arguments

seems to be _necessary. Since we also need the finiteness of

the _group of

_unity

of the

_ring

of

_integers

under

_{consideration,}

we confine

ourselves to the Gaussian case.

Again

our results are derived from a result of the

_type

(2).

The number

field version of

₍₂₎

is established in Thuswaldner

_[18].

We

_only

need the "Gaussian" case of this result: Let b = -~c ~ i be the base of a canonical

number

_system

in with minimal

_polynomial

We define the sets

Then,

for _{m, r E}

_Z,

a E

_{Z [I]}

and an ideal s of

Z [i],

satisfying

we have the estimate

A 1 is an

effectively computable

constant

_independent

of _{r, a, s,} and N. In Section 2 we will

_provide

the _necessary tools needed in the

proofs

of our

_theorems,

Section 3 is devoted to the

_{generalization}

of

₍₃₎

and in

Section 4 the Erd6s-Kac

_type

result

₍₄₎

is established for canonical number

systems

in

_{Z[I] .}

2. AUXILIARY RESULTS

In this section we want to list various results from

algebraic

number

theory,

that will be needed further on. First we

give

some estimates for

sums over

_prime

ideals of

_(in

_fact,

these estimates remain valid for

_rings

of

_integers

of

_arbitrary

number

_fields).

In Narkiewicz

_[17,

Lemmas

_7.3-7.6]

The sum is extended over all

_{prime ideals p}

whose norm is less than x.

Recall Abel’s

_identity

_(cf.

_{Hardy, Wright}

_[8,

Theorem

_421])

for

_{A(t) =}

_an and a

_continuously

differentiable function

f (t).

Apply-ing

this to

₍₇₎

_yields

the estimate

Furthermore,

we have

_(cf.

Narkiewicz

_[17,

_Chapter

_7,

Exercise

_7])

We will also need the estimate

which can be

_proved

in a similar _way as Theorem 429 in

[8].

Next we recall a result due to Hua

_[9]

on

exponential

sums over number

fields. Let

_tr(z)

denote the trace of z

_{over Q}

and let D =

2Z[i]

be the

different of

_Q(i).

_{Then, writing}

_e(x)

=

exp(27rix),

-where

runs over a

complete

residue

_system

in modulo

c-1.

In order to _prove our results we will need a version of

Selberg’s

sieve

method and a

_generalized

Kubilius model in number fields. These

_objects

are discussed in Kubilius

_[14,

_Chapter

_X]

_(cf.

also Danilov

_[2]

for related

results).

In a more

_{general setting they}

are studied in

Juskys

[10]

and

Zhang

[20].

Moreover,

Thuswaldner

_[19]

_gives

a _survey on

Selberg’s

sieve

and the Kubilius model in number fields. We start with the statement of a

quantitative

version of

_Selberg’s

sieve in

_(cf.

_[19]).

Lemma 2.1. Let an E

(1 n N)

and p &#x3E; 0. _{Let ~~, ... , p, be}

prime

ideals

_of

with ... _p and set 11 = _{pi -" p,.}

is an ideal that divides il

(~~~

then we assume that there exists a

representation

where X and R are real

numbers,

X &#x3E;

_0,

and =

for

divisors

_{of fl}

with

_(Dl,

_D2)

= ₀

(o

denotes the unit

ideal).

Furthermore

for

each

_prime

ideal _p we assume that 0

17(P)

1,

holds.

Set

Then the estimate

holds

_{uniformly for}

_p &#x3E;

_2,

w(c~)

is the number

_of

distinct

_prime

ideal

_{factors of -0,}

and

Now we sketch the construction of the

_generalized

Kubilius model for

Z[I].

Again

we refer to

_[19]

for details. Let the same notation as in

Lemma 2.1 be in force. Set

and,

for tlû,

. I r

Let A be the set

_{algebra generated by}

the sets

Et.

In order to make A to a

probability

_space, we define the measure

If all conditions for the

_application

of Lemma 2.1 are

_fulfilled,

it can be

with H as in Lemma

_2.1,

1 Oll [

1 and

₈₂

1. Here

_{[t, cJ}

denotes the least

common

_multiple

of t and D.

Now we set

for each

_{Unfortunately, pEt}

:= _~~ in

_general

does not define a

proba-bility

measure on

,A,

because there _may exist

_empty

sets

_Et

with

_positive

pEt.

Therefore we have to use a trick due to

_Zhang

_[20]

in order to construct a _sequence of

independent

random variables such that the distribution of

their sum has

_density

_ut.

Remark 2.1. The fact that

_pEt

in

_general

does not define a measure on

,A was observed

_{by Zhang}

_[20].

_{Unfortunately,}

_many versions of Kubilius models

_proposed

so

_far,

treat

_pEt

as a measure

(cf.

for instance

[14, 3, 19]).

This lacuna can be mended in _many cases with

_help

of a trick used in

Zhang

[20]

(cf.

also the

_proof

of Lemma 2.3 of the

_present

_paper).

We

want to mention

_explicitely,

that the Erd6s-Kac

_type

theorems in

Mauduit-S6xk6zy

[15, 16]

are not affected

by

this lacuna. Their

proofs

can be

adapted

by reasoning

in the same _way as in the

_present

_paper.

Lemma 2.2.

_Having

the same

_assumptions

as in Lemma

_~.1,

_suppose, that

R(N, t)

=

0 (NA’)

and T = such that

A’+

7J 1. Then

Et :A 0

for

and N

_{large enough.}

Proof. Using

(10)

we

_easily

obtain

On the other

_{hand, by}

the

_assumptions

in the statement of the

_present

lemma we have

Both _{constants cl, c2} are absolute.

By

the estimate

(13)

the result follows.

0 After these

_preparations

we establish the

following

result

(cf.

also

Elliot

_[3,

Lemma

_3.5]).

Lemma 2.3.

_Having

the same

assumptions

as in Lemma

_2.1,

_suppose that

for

N(~)

T~

for

some

/3

&#x3E; 0.

_Define

the

_strongly

additive

_function

where

_l(p)

is a

function from

to R.

Define

the

independent

random

variables

_Wp

_for

each

_prime

_{divisor p}

_{of 11 by}

Then

do not

_{differ by}

more than

/ ,

where

_{3’

=

min( 1, (3).

This holds

_{uniformly for}

all sets F.

Proof.

Consider the variables

It is easy to see, that

_{P(E Wp}

E

_{F) _ p(n : g(an)}

E

_F).

Thus it remains

to show

_{that it}

and v

_approximate

each other.

Let E be the set

_{of all () 111,}

for which

_E

0.

Then v is

_again

a

probability

measure on the set

_{algebra generated by}

the elements of S.

Following

Thuswaldner

_[19,

_p.

_496]

we

_get

the estimate

It remains to treat the sets

_Et

with

_N(t)

&#x3E; T. We

_get

from

_[19,

Lemma

_1]

(cf.

also

_[3,

Lemma

_2.3])

By

our

_assumption

we have

Applying

(15)

to the sums

_containing

_{put in}

(16),

we arrive at

(14)

(15)

and

₍₁₇₎

now

_yield

the result. D

3. PRIME POWERS

This section is devoted to the

_{generalization}

of Gelfond’s result

_(3).

In-stead of the

_ordinary

zeta function in our theorem the Dedekind zeta

func-tion of the number field

_Q(i)

occurs.

Theorem 3.1.

_Suppose

_m, r E Z

_fulfill

₍₅₎

_for

a base b

of

a canonical

number

_system

in

_Z[i].

Let

_{T6 f(N)}

be the number

_of

elements

_of

_Ur,m(N)

that are not divisible

_by

the

_{f -th}

_power

_of

a

_prime

(f &#x3E; 2).

Then there is a constant _1L 1

independent of

N,

such that the estimate

holds.

Proof.

Define the function

where the s-sums run over all ideals of in the indicated _range.

_Setting

N1

=

N~1-~»2,

, where A is as in

(6),

we

split

the last double sum in two

parts:

For

_{IR21 ]}

we obtain the estimate

Since there are

0(rl) (E

&#x3E;

_0,

arbitrary)

elements k in with

N(k)

= r

(cf.

[17,

Lemma

_4.2]),

we

_get

and,

hence

Keeping

in

_mind,

that

Nl

=

N(1-À)/2,

_we

get

for c small

_enough

In order to extract the main term, we

apply

(6)

to

Ri

to derive

Since

and

Setting p

=

max(al, a2, a3)

1

_yields

the result. 0

4. AN

ERD6s-KAc

TYPE THEOREM

Now we _prove an Erd6s-Kac

_type

theorem for the

_complex

sum of

_digits

function. This extends

₍₄₎

to number

_systems

in

_Z[i].

_Throughout

this

section

_~b(x)

denotes the normal law. Furthermore we use the notation

f (x) C 9O)

for

_{f (z) =}

_0(g(x)).

Theorem 4.1.

_Suppose

_{m, r E} Z

_fulfill

₍₅₎

_for

a base b

of

a canonical

numbers

_system.

in

_~(i).

_Define

the

_frequency

where

_w(z)

counts the distinct

_prime

_{factors of}

z E

_Z[i].

Then

holds

_{uniformly for X}

and N &#x3E; 8.

Proof.

We will construct a _measure, with

_respect

to that the additive

func-tion behaves like a sum of

independent

random variables. To this matter we will

apply

the

general

Kubilius model defined in Section 2. In a

second

_step

we will use the

_Berry-Esseen

Theorem

(cf.

[1, 4])

to show the

convergence to the normal law.

Set t = _exp and

let w, (z)

count the distinct

prime

factors

(

log log log

N

p of z, whose norms

_satisfy

N(b)

N(p)

t.

_LN(X)

shall denote the

frequency emerging

from

_QN(X)

if one

_replaces

_by

wi (z) (We

will also use these functions with ideals as

_arguments.

In this case

they

count the distinct

_prime

ideal factors of their

_argument.

Since is a

_principal

ideal

domain we have

w(z)

= and

wl(z)

=

wl(z7G(il)

for

any z E

_Z[i]).

Now we want to

_apply

the Kubilius model:

_assign

the elements of

to the numbers _aj of the model and set X =

IUr,m(N)I

and 17(q)

=

N(q)-l.

For the

_prime

_{ideals pl, ... , ps of the model} we take all

_prime

ideals p with

N(b) N(p)

t. The function S of the model turns out to be

Using

(8)

we

_get

the estimate S =

log t(1

₊

o(l)).

Hence,

the conditions of

the model allow T to be _{any power} of N. Our first

_goal

is the

_application

of Lemma 2.3. To this matter we have to

_show,

that the sets

_Ee

are

_non-empty

up to a certain value of

N(~).

We do this

_{by using}

Lemma 2.2.

_Hence,

we

have to estimate the remainder terms

_{R(N, t).}

_{It is easy} to see, that in our case the remainder terms

_{R(N, t)}

are

given

by

In order to estimate the

_{R(N, t)}

we use Hua’s result

(11).

Let o be the

different of

_7G~i~.

Then

The sum runs over a

complete

residue

_system

in

mod-ulo

c~-1

not

_containing

the element - 0

_(c~-1).

We will use this notation in

the rest of this _paper.

_Setting

for a E

_yields

In the next

_part

of our

proof

we need the

following

estimates

(cf.

[18,

Lemmas 3.2 and

_3.1]):

For 1 = 0 _we have

Moreover,

for

_{any ~}

E

_Q(i)

we

have,

for :

Using

the estimates

₍₁₈₎

and

₍₁₉₎

_yields

Hence,

with A’ =

max( ) , A)

we have

Set

_11t’

and

By

(20)

and

₍₂₁₎

the conditions for the

_application

of Lemma 2.2 are

ful-filled. Since X = =

O(N)

by

(6),

_we

conclude,

that

Et =,4

0

for

_N(t)

. With that we have established the conditions for

the

_application

of Lemma 2.3. Define a collection of

_independent

random

variables

_by

Lemma 2.3 now

_yields

where £’

indicates,

that the sum runs over all

square-free

ideals

t,

whose

prime

factors p have

N(b)

N(p)

t.

The estimate

₍₂₂₎

holds for all t &#x3E; 2 and for

_{max(log t,}

_{S) 1 8 .}

Some calculations

_yield

that the first error term in

(22)

is for

each d E N.

Now we estimate the second error term. Recall that there are

(6

&#x3E;

_0,

_arbitrary)

ideals in

_{Z [I]}

whose norm is

_equal

to r. Since

I R (N, t)

I

=

The last

_step

is a _consequence of

(6).

This

_implies,

_together

with

_(22),

that

Now we turn to the second

_step

of our

_proof.

In order to

_apply

the

Berry-Esseen Theorem on the _convergence of a sum of

independent

random

vari-ables to the normal law

_(cf.

_{~1, 4~),}

we define the random

variables

_Zp

_by

Let

_Then,

_by

₍₉₎

we have

The

_Berry-Esseen

Theorem now

_{yields, again using}

(9),

For the random variables

_Wp

we derive

Together

with

₍₂₃₎

this

_yields

Since a number z with

Izl2

N has at most

_{log log}

_N)

_prime

factors

p with norms not contained in the interval

_{[N(b), t], we}

can

replace

LN(X )

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J6rg M. THUSWALDNER

Institut fur Mathematik und _Angewandte Geometrie

Abteilung fur Mathematik und Statistik Montanuniversitat Leoben

FYanz-Josef-Strasse 18 A-8700 LEOBEN AUSTRIA