Multiplicative functions and $k$-automatic sequences

S

OROOSH

Y

AZDANI

Multiplicative functions and k-automatic sequences

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{2 (2001),}

p. 651-658

<http://www.numdam.org/item?id=JTNB_2001__13_2_651_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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

Multiplicative

functions

and

k-automatic

sequences

par SOROOSH YAZDANI

RÉSUMÉ. Une suite est dite

_{k-automatique}

si son ne terme peut

être

_engendré

_par une machine à états finis lisant en entrée le

développement

de n en base k. Nous _{prouvons que, pour} de

nom-breuses fonctions

_{multiplicatives f,}

la suite

_(f(n)

mod

_u)n~1

n’est pas

k-automatique.

C’est en

particulier

le cas pour les fonctions

multiplicatives 03C4m(n), 03C3m(n), 03BC(n)

et

_~(n).

ABSTRACT. A sequence is called k-automatic if the n’th term in

the sequence can be

_{generated by}

a finite state

_machine,

_reading

n in base k as _input. We show that for _many

_{multiplicative}

func-tions, the sequence

(f(n)

mod

u)n~1

is not k-automatic.

_Among

these

_{multiplicative}

functions are

03C4m(n),

03C3m(n),

03BC(n),

and

~(n).

We call a function

f :

N B

101

- C

_{multiplicative,}

if for all _{m, n} E

N B

101,

m and n

coprime,

we have

f (mn)

=

f (m) f (n).

As usual let

T(n),

Q(n), ~(n),

represent

the number of divisors of _n, sum of the divi-sors of _n, number of numbers less than or

_equal

to n and

prime

to n,

and the M6bius function

_{respectively.}

We know that

_T(~),

_{Q(n), ~(n),}

and

_/-L(n)

are

_{multiplicative.}

Also let be number of elements in

{ ~au a2~ ... a~)

I

alaz ... am = n and

al,a2,... ,am E

_{N‘ ~}

~0~ ~.

Then

we have

where

_pi’s

are distinct

_primes

(see

for

example

[9,

_p.

72]).

Furthermore let

Recall that is

_{multiplicative}

for all

_integers

m. Note that

al(n) =

a (n),

and

_T2(n)

=

T(n).

Given k &#x3E;

_2,

we _say a _sequence T = is k-automatic if and

only

if

-is

_finite,

where =

(t(kln

₊

r))~,&#x3E;1.

The set

T~k~

is called the k-kernel

of T. We _say a set S C

_{I~Y B 101}

is k-automatic if the _sequence is

_k-automatic,

where

-If t :

_{I~Y B}

₁₀₁

- X for some set

_X,

and if there is a

_{mapping -D :}

X -

Y,

then we can extend $ to sequences in X with

~(T) =

(~(t(n))n&#x3E;1).

Note that

is an onto

_{mapping. Specifically}

note that the

_cardinality

of

T~~&#x3E;

is

_greater

than or

_equal

to the

_cardinality

of

_~(T)~k~,

and hence if T is

_k-automatic,

then so is

~(T).

Therefore we have the

following,

Lemma 1. Let be a _sequence

of

_integers.

If

there exist

_integers

v, k &#x3E;

2 such that

_{( f (n)}

mod

_V)n&#x3E;l

is

_k-automatic,

then

_for

all

_qlv

we have

that the _sequence

_{( f (n)}

mod is also k-automatic.

The term k-automatic is used because one can

_compute

t(n)

by feeding

the base k

_{representation}

of n as an

_input

to a finite state machine

_[5].

In

[3],

see also

[4],

it is shown that

_given

prime

_p and a _sequence

with values in

_IEP,

then

is

_algebraic

over

_{Fp (X)}

if and

_only

if is

_p-automatic.

Now we

_proceed

to _prove the first theorem in this _paper, whose

_proof

is

a variation of a

proof suggested

by

J. Shallit.

Theorem 2. Let v &#x3E; 1 be an

_integer

and

_f

a

multiplicative function.

As-sume that

for

some

_integer

h &#x3E; 1 there exist

infinitely

_many

primes

_ql

such that - 0

_{(mod v).}

Furthermore assume that there exist

rela-tively

prime integers

b and c such that

for

all

_primes

_q2 = c

_{(mod b)}

we

have

_{l(q2) t}

0

_{(mod v).}

Then the _sequence F =

( f (n)

mod is not k-automatic

_{for any k &#x3E;}

2.

Proof.

Choose an

_arbitrary

integer k &#x3E;

2. Since

gcd(b, c)

=

1,

by

Dirichlet’s theorem there exists some

_integer

m such that bm + c &#x3E;

_k,

and bm + c is

prime.

Letting a

= bm ₊

Now we will show that

_{given l, rl, r2}

E

_{N ) (0)}

such

that kl

&#x3E;

_2bk,

0

r2

kl,

and r2 - a

(mod bk),

there exists n E

_{1‘~ B}

(0)

such that

¡(kin

+ +

_{r2) (mod}

_v),

hence This in turn means

that the k-kernel of F is

_infinite,

which means that F is not k-automatic.

Choose a

prime

_ql &#x3E;

kl

such that 0

_{(mod v).}

Observe that

= 1 since _ql is

prime

and _{ql &#x3E;}

k~

_&#x3E; b. Hence there exists an

integer

no such that

Furthermore observe that

for all no. Therefore for

all j

E

N B {0},

we have

and

We need to show that for some

j,

no +

jq’4+lb

&#x3E; 0 and the left-hand side of

_Equation

₍₃₎

is

_prime.

To do so we will show that

gcd(nobkl

+

1,

and

_apply

Dirichlet’s theorem.

Note that r2 - a

(mod k),

and

gcd(a, k)

= 1. Therefore

Also

_nobkl

+ r2 - a

(mod b).

Since

gcd(a,

b)

=

1,

we

_get

Finally

from

_Equation

₍₂₎

we have that &#x3E;

We know that _rl

_,-E

_r2, and I

that

. Since ql is

prime,

we

get

Therefore

_gcd(nobkl

+ r2,

klq?+lb2)

= 1. Hence

by

Dirichlet’s

theorem,

we can find an

_{integer j}

&#x3E;

_I

such that

is

_prime.

_{By hypothesis,}

+

_bno)

+

_r2)

_{mod v =f.}

0. On the other hand we have that

by

Equation

(2)

Since

_f

is

_{multiplicative,}

we have

Therefore F is not k-automatic for _any k &#x3E; 2. 0 From this theorem we

_immediately

_get

the

following

corollaries.

Corollary

3. Given m &#x3E; 1 and v &#x3E;

_3,

the _sequence mod

not k-automatic

_for

_any k &#x3E; 2.

Proof.

Given an

_{integer v &#x3E;}

_3,

there are

infinitely

_many

_primes

_{ql - 1}

(mod

v).

Taking h

= v - 1 we

_get

- __...._

Also for

_primes

q2 - 1

(mod v),

we have

am(q2)

mod v =

_2,

since _{v &#x3E;} 3. So the

hypotheses

of Theorem 2 are

satisfied,

and hence mod

is not k-automatic. D

Corollary

3 answers the

_question

raised

_by

Allouche and Thakur of whether

is

_always

transcendental over

Fp (X)

for odd

_primes

_p

[2].

_{They proved}

the

transcendence of

_Equation

₍₄₎

for _many cases of _p and m in order to

_give

a

proof

of the function field

_analogue

of Mahler-Manin

_conjecture.

Since

(am(n)

mod

_P)n&#x3E;1

is not

_p-automatic

for

_{primes p &#x3E;}

_3,

_using

Christol’s theorem

_[3]

and

_[4]

we

_get

that the formal _power series in

Equation

(4)

is

_always

transcendental over

-Corollary

4. Given v &#x3E;

_3,

the sequence

(§(n)

mod is not k-automa-tic

_for

_any k &#x3E; 2.

Proof.

Note that

Hence

_given

a

_prime

_{ql -} 1

(mod v)

we have

O(ql) =-

0

(mod v).

Also,

given

a

_prime

_{q2 = -1}

_{(mod v)}

we have

~(q2) -

-2

(mod v).

Since v &#x3E; 3

the

_hypotheses

of Theorem 2 are satisfied. Hence is not

Note that

_(4)(n)

mod

_2),~~1

is k-automatic for all

_k,

since

_0(n)

is even

for all n &#x3E;

_2,

and hence

_(~(n)

mod

_2)n&#x3E;l

is constant for n &#x3E; 2.

We also

_get

the

_following

well-known

_result,

which is a direct _consequence

of the fact that

_square-free

numbers are not k-automatic

_[5,

_p.

_183].

Corollary

5. Given an

_{integer v &#x3E; 2,}

the _sequence mod is not

k-automatic

_for

_any k &#x3E; 2.

Proof.

This is a direct _consequence of Theorem 2. _ D

The

_proof

of Theorem 2 relied

_heavily

on the existence of

_primes

_q such

that

_(mod

_v).

Now we look at another set of

_{multiplicative}

func-tions

_{f ,}

where for all

_primes

_q, and some

_integer

v. The

_technique

used in this section is different from that used in the

_proof

of the

_previous

theorem,

and we need to

_give

the

_following

definition.

Definition 1. Let T = and

#S

be the number of elements in the set S. Then the

_density

of the

_symbol

a in the _{sequence T} is defined to

be ,, , ~ , .... ,

if the limit

_exists,

and is undefined otherwise.

Using

this definition we will cite the

_following

lemma due to

_Minsky

and

Papert

[8],

see also

[5,

_p.

184].

Lemma 6. For _any k-automatic _sequence

_F,

_if

_{d(F, a)}

= 0 then

where

_{aj is}

the

_position

_{of the j’th}

occurrence

of

a.

We now

_proceed

to _prove the

_following

theorem.

Theorem 7. Let v &#x3E; 1 be an

integer,

and let

f

be a

_{multiplicative function}

such that

_for

some

function

_g, where the _pi are distinct

primes.

Also _suppose that

_{g(l) =-}

0

_{(mod v)}

and that there exists some

integer h &#x3E;

1 such that

_{g(h) 0}

0

_{(mod v).}

Then F =

( f (n)

mod

not k-autorraatic

_for

_any

_{integer k &#x3E;}

2.

Proof.

First,

we need the

_following

lemma.

Lemma 8. Let

_f

be a

multiplicative function

such that

f (q) -

0

(mod v)

for

all

_primes

_q. Then

Proof.

Note that if 0

_{(mod v),}

then n is a

_powerful

number

(a

number where each of its

_prime

factor occurs to a _power

_greater

than

1).

Choose

a ~

0

_{(mod v).}

From

_[6],

see also

[7,

_p.

178],

we have that for _any

e &#x3E; 0

for

_{large enough}

n.

_{Choosing E}

I

we

_get

that

d(F,

a) =

0 for 0. Hence

the desired result follows. l7

Now we are

_ready

_{to prove} our next Theorem. Let a =

g(h)

mod v and

aj be the

j’th

occurrence of a in F.

_By

definition of

_h,

we

_get

_{a ~}

0. From

Lemma 8 we have

d(F, a)

= 0. So if we show that

_{lim supj,,,.}

a

=

1,

_we

aj

are done.

On the other

_hand,

note that is a

subsequence

of where

pi is the i’th

prime.

Therefore for

all j

there exists i such that

Hence

Therefore

But

_limsupi,

=

limi

=

1,

this is _an immediate _consequence of the

_prime

number theorem

_limi,.

_Pi

_/i

_log

i = 1.

Therefore 1. It follows that F is not k-automatic

for

_{any k &#x3E;}

2. 0

Corollary

9. Given an

_integer

m &#x3E;

_1,

the _sequence mod

not k-automatic

_for

_any k &#x3E; 2.

Proof.

Let n =

2Q:d,

where d is odd. Then we have

Furthermore,

we know that

T(d)

is odd

only

when d is a

_perfect

_square.

So mod 2 = 1 if and

only

if n is _a

perfect

square times a _power of 2.

On the other hand if an

represents

the

_position

of the n’th occurrence of

1,

we

_get

So we have

(am(n)

mod

2)n&#x3E;l

is not k-automatic

_by

Theorem 7. 0 Also

_combining

Theorems 1 and 2 we

_get

the

following

new result.

Corollary

10. For all

_integers

_{v, rrz,} k &#x3E; 2 the sequences

(Tm(n)

mod

V)n2:1

is not k-automatic.

Proof.

Assume that for some

v, m, k &#x3E;

2 the _sequence mod

v)n&#x3E;1

is k-automatic. Therefore

_by

Lemma 1 we

_get

_given

an

integer

p~v

the

sequence

(T m (n)

is also k-automatic. Therefore assume without

loss of

_generality

that v is a

prime.

Consider the

following

cases.

Case 1:

_{m ~}

0

_(mod

_v).

Then if we choose a and h such that va

11

m-1

and h - 1 - m

(mod

va+i)

we

_get

Therefore for any

prime q

we have mod v = 0

by

(1).

On the other hand for all

_{primes q}

we have mod v = 0. Hence

by

Theorem

_2,

we

_get

that mod is not k-automatic.

Case 2: m - 0

(mod v).

Let

g(h) =

We have that

/(TIpiQi) =

TIg(ai)

by

(1).

Also we know

g(1)

= m - 0

(mod v).

Assume that

va 11m.

Then we

_get

g(va) t

0

(mod v).

Therefore

by

Theorem

7,

mod

is not k-automatic. D

Corollary

10 can be used to prove the transcendence of xq

(an

analogue

of

x in the field

GF(q)((X)) )

over the field

GF(q)(X)

[1].

It is worth

_mentioning

that both of our theorems relied on

vlf (n),

for some n. If

_f

is

_{multiplicative}

and v t

f (n)

for

_{any n &#x3E; 1,}

then its the

analysis

becomes much more difficult. For

example

the Liouville function

defined

_by

is never divisible

_by

_any

_prime.

It seems that the

_question

of whether or

not is k-automatic is an _open

problem

worth

pursuing.

Acknowledgement.

I would like to thank J. Shallit for

_simplifying

the

proof

of Theorem

_2,

and

_suggesting

different

_{generalizations}

of the

_original

the referee for _many valuable

_suggestions,

and P.

_Borwein,

G.

Almkvist,

and W. L. Yee for all their

_help

in the

_preparation

of the

_original

draft. References

[1] J.-P. ALLOUCHE, Sur la transcendance de la série _formelle 03C0. Sém. Théor. Nombres

Bor-deaux 2 _(1990), 103-117.

[2] J.-P. ALLOUCHE, D. S. _THAKUR, Automata and transcendence _of the Tate _period in _finite

characteristic. Proc. Amer. Math. Soc. 127 _(1999), 1309-1312.

[3] G. _CHRISTOL, Ensembles presque periodiques k-reconnaissables. Theoret. Comput. Sci. 9

(1979), 141-145.

[4] G. CHRISTOL, T. KAMAE, M. MENDÈS FRANCE, G. RAUZY, Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108 _(1980), 401-419.

[5] A. _{COBHAM, Uniform tag sequences.} Math. _{Systems Theory} 6 _(1972), 164-192.

[6] P. _ERDÖS, G. SZEKERES, Über die Anzahl der Abelschen _{Gruppen gegebener Ordnung} und über ein verwandtes zahlentheoretisches Problem. Acta Sci. Math. _(Szeged) 7 _(1935), 95-102.

[7] A. P. _SHIU, The distribution _{of powerful integers.} Illinois J. Math. 26 _(1982), 576-590.

[8] M. _MINSKY, S. _{PAPERT, Unrecognizable} sets _of numbers. J. Assoc. _Comput. Mach. 13 _(1966),

281-286.

[9] R. _{SIVARAMAKRISHNAN,} Classical _{theory of} arithmetic _{functions. Monographs} and Textbooks

in Pure and _Applied Mathematics _126, Marcel _{Dekker, Inc.,} New York, 1989. Soroosh YAZDANI Department of Mathematics University of Illinois 273 _{Altgelt Hall,} MC-382 1409 W. Green Street Urbana, IL 61801 USA