On the number of subgroups of finite abelian groups

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

LEKSANDAR

I

VI ´

C

On the number of subgroups of finite abelian groups

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{2 (1997),}

p. 371-381

<http://www.numdam.org/item?id=JTNB_1997__9_2_371_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

On the number of

_subgroups

of finite abelian groups

par ALEKSANDAR

RÉSUMÉ. Soit

T(x) = _K1x

log2

_{x + K2x logx + K3x +}

_0394(x),

où _T(x) désigne le nombre de sous _groupes des _groupes abéliens dont l’ordre n’excède _{pas x} et dont le rang n’excède pas 2, et _0394(x) est le terme d’erreur. On démontre que

03942(x)

dx ~

X2 log31/3

X, dx

_03A9(X2log4

ABSTRACT. Let

T(x) = _K1x

log2 x +

_{K2x log x + K3x + 0394(x),}

where _T(x) denotes the number of subgroups of all Abelian groups whose

order does not exceed x and whose rank does not exceed 2, and _0394(x) is the error term. It is proved that

1. Introduction

Let

where

_T(G)

denotes the number of

_subgroups

of a finite Abelian _group

Q,

is the rank

_{of 9,}

and

₁₉

is the order of

Q.

The

_{group g}

has rank r

if

1991 Mathematics _{Subject Classification. 11N45, 11L07, 20K01, 20K27.} Manuscrit _requ le 5 _juin 1997

so that one has

where

_Kj

are effective constants and

_0(x)

is to be considered as the error

term in the

_asymptotic

formula for

_T(x).

One has the Dirichlet series

representation

(this

is due to G. Bhowmik

_[1];

the

_generating

Dirichlet series for Abelian _groups of rank &#x3E; 3 are more

complicated)

Using

(1.2)

and the estimate in the four-dimensional

_asymmetric

divisor

problem

of

_H.-Q.

Liu

_[6],

G. Bhowmik and H. Menzer

_[2]

obtained the bound

with c =

31/43

= 0.72093....

Recently

H. Menzer

[6]

used two new

esti-mates in the three-dimensional

_asymmetric

divisor

_problem

to _prove

_(1.3)

with the better value c =

9/14

=

0.64285... ,

and this is further

improved

in

the

_forthcoming

_paper

_by

G. Bhowmik and J. Wu

_[3]

to

_0(x)

C

~5~$

log4

~.

Note that we can write

(1.2)

as

where the Dirichlet series for

_U(s)

is

_absolutely

_convergent

for Vie &#x3E;

_1/3.

This

_prompts

one to think that in

_(1.1)

there should be a new main term

corresponding

to the

_pole

of order 3 of

_H(s)

at s =

1/2,

namely

that _we

where 0

_(one

cannot

_hope

for

_E(x)

=

o(xl/2)

since in

[3]

it was shown

that

_E(x)

=

O(Xl/2(log log x)6)

holds).

Even if the relation

(1.5)

is

perhaps

too

_optimistic,

_{it is very}

_likely

that

_0(x)

G

_holds,

and that

_A(x)

cannot be of order lower than

x 1/2

_log2

x. In fact H. Menzer

[5]

_conjectured

that

This was

_{proved by}

Bhowmik and Wu

[3],

which is a

_corollary

of their

bound __

Since

_{heuristically}

in

_(1.5)

the terms +

C2 log x

+

_C3)

are the

residue of

_H(s)x8/s

at s =

1/2

it is _not difficult _{to see} that the _constant

Gl

is

_negative,

so

actually

in

(1.6)

Q is

S2_,

i.e. the S2-result of Bhowmik

and Wu is , ,

The

_object

of this note is to

_investigate

_A(x)

in mean _square, and we shall

prove two

fairly

precise

results contained in

THEOREM 1. We have

THEOREM 2. We have

Remark 1. The

_omega-result

_(1.8)

_implies

another

_proof

of Menzer’s

conjecture

(1.6).

Remark 2. It is

_plausible

to

_{conjecture that,}

for X --~ oo and suitable

C &#x3E;

_0,

one has

--although

this seems to be out of reach at

_present.

Remark 3. It will be clear from the

_proof

of Theorem 2 that the method is

_capable

of

_{generalization}

to the case where the error term in

_question

corresponds

to the Dirichlet series

_{generated by}

suitable factors of the form

~(as

+

_b)

_(a,

b

_integers).

2. Proof of the _upper bound estimate

To _prove Theorem 1 we start from the relation

where c &#x3E; 0 is a suitable constant. The formula

_(2.1)

follows from the

properties

of Mellin

_{transforms, similarly}

as in the case of the classical

divisor

_problem

_(see

_(13.23)

on _p. 357 of

[4]).

If the

_integral

on the

left-hand side of

_(2.1)

_converges, so does the

_integral

on the

right-hand

side and

conversely.

We shall need the

_following

facts about

_{C(s) (see}

_[4]

for

_proofs):

The last bound follows _e.g. from Th. 4.4 and Th. 5.2 of

_[4].

Now we take c =

1/2

₊

1/ log X,

X &#x3E;

Xo

_&#x3E; 0. Then from

(1.4)

and

(2.1)

_we obtain first

By

symmetry

we have

say. Since

C(s)

C

_{1/ls - 1~ [}

near s =

1,

we have

₁₁

W

log X.

Using

(2.2)

it follows that

_(Ci

&#x3E; 0 is a

constant)

Let

By

integration

by

parts

it follows that

But we have

with the

_change

of variable v =

log

t/ log

X. We also have

Therefore

and

_(2.3)

_gives

Replacing

X

_by

E N and

_summing

the

_resulting

estimates we

obtain

_(1.7).

3. Proof of the

_omega-result

To prove the

omega-result

of Theorem 2 we shall use the method used in

proving

Theorem 2 of

_[5],

with the _necessary modifications.

_Namely

in

_[5]

the

_generating

Dirichlet series was of the form

where 1

a2

... ak are

_integers,

with k

possibly

infinite

(e.g

the

_generating

series of the function

_a(n),

the number of

_{non-isomorphic}

Abelian _groups with n

_elements,

is

_(3.1)

_{with aj}

_{= j,}

k =

oo).

The Dirichlet

series

_H(s)

_(see

_(1.4))

is

_clearly

not of the form

_(3.1),

since it contains the factor

_{C(2s - 1).}

_Writing

where

_d(n)

is the number of divisors of n. We remark that Bhowmik and

Wu

_[3]

_proved

the

_sharper

bound

_t2(n)

G but for our

purposes

(3.2)

is more than sufficient. As on _p. 82 of

_[5]

we start from the

Mellin inversion

_integral

_(see

also _p. 122 of

_[4])

where h,

U &#x3E; 0. We shall take

T 1-~

t

_{T, h}

=

log2

_{T, s}

= 2 +it,

Y =

TB,

where 6 &#x3E; 0 is a

sufficiently

small constant and B &#x3E; 1 is a suitable constant.

Setting

U =

n/Y

we obtain from

(3.3),

_by

termwise

_integration,

We shift the line of

_integration

to =

-1/4

and

apply

the residue

theorem. The

_pole

w = 0 of the

integrand

_gives

the residue

H(s),

while the

poles

of

_{H (s + w)}

_give

a total contribution which is

0 ( 1 )

in view of

Stirling’s

formula for the

_{gamma-function.}

The

_{integral along}

the line 9Bew

_{= - 1/4}

is

_bounded,

and we obtain

The idea of

_proof

is as follows. We shall _prove that

and then use

(3.4)

to show that

_(3.5)

_gives

a contradiction if we assume

that

_(1.8)

does not

_{hold, namely}

that we have

since

_(3.7)

_gives

(

Using

(1.4)

and

_(2.2)

we have

Now let

_F(s)

:=

(2(S)(4(2s)

and use a

general

lower bound for mean values

of Dirichlet series

_(see

e.g. K. Ramachandra

[8], [9];

note that the factor

1/n

is

_missing

in

_(4.2)

of

_[5]):

(3.8)

where

_F(s)

=

1+~~=2

c(n)n-S

_converges for 91:e = Q &#x3E; _uo,

F(s)

is

regular

for

9te ~

_1/2,

M t

2M and both

_F(s)

«

eMD

and

_c(n)

«

MD

hold for some D &#x3E; 0. In our case

where the divisor function

_d4(n)

is

_{generated by}

_~4(s).

Hence

_(3.8)

_yields

by partial

summation from

_{Cx log3 x}

_(C

&#x3E;

_0).

Thus

_(3.5)

is

_proved,

and it remains to

see how

it leads to the

_proof

of Theorem 2.

To obtain the left-hand side of

_(3.5)

from

_(3.4),

we shall divide

(3.4)

by t,

square and

integrate

over t T. We use the mean value theorem

for Dirichlet

_polynomials

_(see

Theorem 5.2 of

_[4])

to deduce that

for 6

_{sufficiently small,}

where we used the bound

(3.2)

and the trivial bound

It remains to evaluate

This is done

_{again by}

the use of the mean value theorem for Dirichlet

polynomials.

However first we

_{integrate by}

_parts

and use

(1.1)

to obtain

say.

By

the first derivative test

(Lemma

2.1 of

[4])

it is seen that

11

W

for B 2 -

_~8.

In

₁₂

we use

integration by

_parts

and the bound

0(x)

C

x 211

_(see

_(1.3))

to obtain

v y

The contribution of the 0-term to I will be

_negligible,

and so will be also

the contribution of

_h(x/y)h

+

_1/2

if B 3 - 38. The main contribution

to I from

12

will be from the term it. This is

By using

the

_{Cauchy-Schwarz inequality}

for

_integrals

and

_inverting

the order of

_integration

it is seen that the last

_integral

does not exceed

where we used

_again

the mean value theorem for Dirichlet

_polynomials.

If

(3.6)

holds,

then

_obviously

also

which is the contradiction that _proves Theorem 2.

REFERENCES

[1]

G. Bhowmik, Average order of certain _functions connected with arithmetic _of ma-trices, J. Indian Math. Soc. 59 _(1993), 97-106.

[2]

G. Bhowmik and H. _Menzer, On the number _{of subgroups of finite} Abelian _groups, Abh. Math. Sem. Univ. _Hamburg, in _press.

[3]

G. Bhowmik and J. _Wu, On the _asymptotic behaviour _of the number of subgroups of finite abelian groups, Archiv der Mathematik 69 _(1997), 95-104.

[4]

A. The Riemann _{zeta-function,} John _{Wiley &#x26; Sons ,} New York

_(1985).

[5]

A. The _general divisor _problem, J. Number _Theory 27 _(1987), 73-91.

[6]

H.-Q. Liu, Divisor _{problems of 4} and 3 _dimensions, Acta Arith. 73

_(1995),

249-269.

[7]

H. _Menzer, On the number _{of subgroups of finite} Abelian groups, Proc. Conf.

Ana-lytic and _Elementary Number _{Theory (Vienna, July} 18-20, 1996), Universität Wien &#x26; Universität für _Bodenkultur, Eds. W.G. Nowak and J. Schoißengeier, Wien _(1996),

181-188.

[8]

K. _{Ramachandra, Progress} towards a _conjecture on the mean value of Titchmarsh series, Recent _Progress in _Analytic Number _Theory, Academic _Press, London 1 (1981), 303-318.

[9]

K. _Ramachandra, On the Mean- Value and _{Omega-Theorems for} the Riemann

zeta-function, Tata Institute of Fund. _{Research, Bombay,} 1995.

Aleksandar IVI6

Katedra Matematike RGF-A Universiteta u _Beogradu Dju‘sina 7, 11000 _Beograd

SERBIA

_(YUGOSLAVIA)

e-mail : _{aleks~ivic.matf.bg.ac.yu}