On the average number of direct factors of a finite abelian group (“sur les variété abéliennes”)

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

H

ARTMUT

M

ENZER

On the average number of direct factors of a finite

abelian group (“sur les variété abéliennes”)

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{1 (1995),}

p. 155-164

<http://www.numdam.org/item?id=JTNB_1995__7_1_155_0>

© Université Bordeaux 1, 1995, tous droits réservés.

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

On

the

_average

number of direct

factors of

a

finite Abelian

Group

sur

les variété abéliennes

par Hartmut MENZER

1. Introduction

Let

_a(n)

denote the number of

non-isomorphic

Abelian _groups with n elements. This is a well-known

multiplicative

function such that =

’(a).

Historically,

the

_summatory

function

_A(x)

:=

E

a(n)

was first

in-vestigated by

Erd6s-Szekeres

_[2]

in

_1935,

and from that time much research

was done on this

_subject

(for

an account the reader is referred to

_Chapter

14 of A.Ivi6

_[3]

or

Chapter

7 of E. Kratzel

[5]).We

remark,

that the best

published

estimate of

_A(x)

is due to

_H.-Q.

LIU

_[8].

He obtained the result

with

_A(x)

_«~

x40/159+e and

_40/159

= 0.251572.... For _a =

Re(s)

_&#x3E; 1 _we

define the Dirichlet series

00

and

It is easy to show that

where

_C(s)

as usual denotes the Riemann zeta-function.

Furthermore,

we define the

multiplicative

functions

t(n)

and

w(n)

by

the Dirichlet convolution in the

_following

_way

_{t(n) := E a(u)a(v)}

and

uv=n

w(n) :=

E

uv=n

Thus for any

prime

p and

integer

a &#x3E;

1 one has

J2013~

In

_particular,

since

_P(1)

=

1, P(2)

=

2, P(3)

=

3, P(4)

=

5,

_we have

t(p)

=

_2,

t(p2)

=

_5,

t(p3)

=

_10,

t(p4)

= 20 and

w(p)

=

3~Wp2)

=

_9,

22,

W (p4)

= 68.

In this _paper we shall be concerned with

obtaining

estimates for the sums

T(x) = ¿ t(n)

and

W(x) = ¿ w(n).

The

_asymptotic

behaviour

nx

of T(x)

was first studied

by

E. Cohen

[1]

and Kratzel

[6].

It is known that

T(z) = £

T(G),

where

_T(G)

denotes the number of direct factors of an

Abelian _group G.

In

_[10]

Seibold and the author

_proved

the

_{representation}

T(x)

=

with _CE

x45/109+e.

and

_45/109

= 0.412844.... Furthermore A.

Ivi6

_[4]

_proved

several estimates for the

_{sums E}

_{t~‘ (n)}

_{and E}

_{t(nl )}

with

~,~

k =

_{2, 3, 4}

and 1 = 2.

--The aim of this paper is to establish new

_asymptotic

results for

_T(x)

and

_W(x).

_First,

we _prove a

sharper

result for

Ll1(x)

and

_get

the estimate

C

X9/22 log4 X

_(9/22

=

0.409).

Second,

_we establish _a

with the error term

_{A2 (z) «}

X 76/153

_(76/153

=

0.496732 ... ) .

Here

.K2 (x)

are well-known functions which will be defined

by

(32)

to

(37)

later.

The _paper has the

_following

structure. In section two we formulate four

preliminary

lemmas. In section three we _prove four theorems. Theorem 1

and Theorem 2 describe results for two

_special

divisor

_problems

with the dimensions four and

six,

respectively.

In Theorems 3 and 4 we obtain two results based on the first and second theorem.

2.

_Preliminary

Results

Throughout

the _paper Landau’s

_0-symbol

and

_{Vinogradoff’s}

«-notation

(all

constants involved are absolute

ones)

and the well known function

9(t) = t - [t]

-

1/2

are used.

_Furthermore,

we denote the Euler’s

con-stant

_by

_C,

and the constant

₀₁

_by

LEMMA 1.

(1) a) "

Let

_{d(1,1, 2, 2; n)}

denote the divisor

_function

_{d(1,1, 2, 2; n) _}

#{(nl,... n4) :

nlrc2n3n4

=

n}

and let

T3(n)

be

defined by

Then

b)

Let

_{d(1,1,1, 2, 2, 2; n~}

denote the divisor

_function

d(1,

1, 1, 2, 2, 2;

n)

=

#{(~1,... n6) :

= n~

Then

Proof.

It is known that

hence

₍₇₎

and

₍₈₎

follows at once.

LEMMA 2

_(Vogts

_[11]).

Let r &#x3E; 2 and _{al, , .. , ar} be reat numbers with

and Ar := ai + ... + ar .

Let

Then

where the main term

_{H(a; x)}

is

_given

_by

For the error term

_{A(a; x)}

we have the

_{representation}

where

_S(u;

_x)

is

_defined

_by

with the conditions

_{o f}

summation

In

₍₁₀₎

the notations u E

_7r(a)

means that u runs over all

_permutations

7r(a)

of a. In the condition of summation the notation means that _{ni ni+l} for ui = _ai,

aj and i

j

and ni ni+l

otherwise.

Remark. The

_{representation}

₍₉₎

for the main term holds if a, a2 ... ar.

However,

in cases of some

_equalities

we can take the limit values

_(see

LEMMA 3

_(Kr6tzel

_[7]).

Let N :=

_{(~,N2,... ,}

=

1,... ,

r -1)

and _{81,82, ... ,}

_positive

numbers. Let

_S(u,

_N;

_x)

be

_defined

by

I ~, I

i

with the condition

_{o f}

summation

For the numbers

Ni

are valid the

inequalities

. . -.

-Assume that

- - ,

for each

_permutations

of

_{(aI, ... ,}

_ar).

Then the estimate

.

holds with

_{N2 ar- I ...}

x for all

permutations

u.

LEMMA 4

_(Menzer,

Krétzel

_[5]).

Let

_S(u,

_N;

_x)

be

_{defined by}

_(12),

then we

have

Remark. The first both formulas

₍₁₃₎

and

₍₁₄₎

was

proved by

the author

by

applying

two results of three-dimensional sums in

[9].

The third formula

(15)

was

_proved

_by

Kretzel

_[5]

_{by applying}

a result of two-dimensional

exponential

sums.

3. Estimates for

_{T (x)}

and

_{W (x).}

According

to the formulas

₍₇₎

and

₍₈₎

of Lemma 1 one can see that in

or-der to establish estimates for

_{T (x)}

and it is necessary to know the

as-ymptotic

behaviour of the

_counting

functions

_{D(1, ~, 2, 2; x)}

and

D(1, l,1, 2, 2, 2; x),

respectively.

Hence,

we _prove first two theorems

THEOREM 1. For the remainder term

_{~(1,1, 2, 2; x)}

holds the estimate

Proof.

We start with the formula

₍₁₀₎

and

_put

a =

(1,1, 2, 2).

This

yields

where

_{S(u; x)}

is defined in the sense of

(11).

Now instead of function

S (u; x)

we take the

special

function which is defined

by

formula

(12).

All sums

_{S (u, N; x),}

where u E

_{7r (1, 1, 2, 2)}

are divided into two subsums

corresponding

to

n2 z

and z _n2, where z is a suitable

value,

which is

defined later. In the first case _n2 &#x3E; z we take formula

(13)

and obtain

_by

using

and Lemma 3 the estimate

for all

_{permutations u}

E

_{7r(l, 1,2,2).}

In the second case n2 z we use the formula

(14).

_By

_applying

and Lemma 3 we

_get

the estimate

for all

_permutations

u E

_{ir (1, 1, 2, 2).}

Now we _compare the estimates

(17)

and

(18)

and

_get

for z the value

THEOREM 2. Let a =

(1,1, l, 2, 2, 2)

and let

Hi (x)

(i

=

l, ... 6)

be

define

by

Then the

_{representation}

ho lds with

where

_S(u;

_x)

is

_{defined by}

₍₁₁₎

zuith r =

6.

Moreover,

zve have

Proof.

We use Lemma 2 with al - a2 = a3 =

1 ,

c~4 = _{a5 = a6 =} 2

and

A6 =

9. We obtain

_{by simple}

calculations for the main term

_H(a-

_x)

6

the

_{representation}

_{H(a; x~ _ ~}

_{Hi (x) -}

_Now,

we

apply

the formula

(15)

of

~

2.=1

Lemma 4 and substitute nl, n2, n3

by

n3, n4, n5. After that we introduce

two new variables of summation _{nl, n2} and sum

_trivially

over them. Hence we

_get

the

following

estimate

and we have

by

_using

Lemma 3

for all

_{permutations u}

E

_{7r(1, 1,1,2,2,2).}

From this estimate it is

_easily

seen that

- -

-Therefore,

we can use the

_inequalities

in

_(27).

Then

and

₍₂₅₎

follows

_immediately.

, THEOREM 3.

_{Let Cl, C2, C3, C4}

be

_defcned

_by

Then

THEOREM 4. Let

_(i

=

l, ... ,

6)

be

defined by

B‘ -

/ J

where

_(i

=

1, ... ,

6)

_are

(19)

_to

(24).

Then

1

Proof.

We

_apply

_equation

₍₈₎

and obtain

with a =

( 1,1, l, 2, 2, 2) .

_Now,

_{we use our} Theorem _2.

It is

_easily

seen that

00 ..

Since the Dirichlet

_series

are

_absolutely

_convergent

for u &#x3E;

_1/3,

REFERENCES

[1]

E. _COHEN, On the average number of direct _{factors of} a _finite Abelian group, Acta

Arith. 6

_(1960),

159-173.

[2]

P. ERDÖS and G. SZEKERES, Öber die Anzahl Abelscher Gruppen gegebener Ordnung und Âber ein verwandtes zahlentheoretisches Problem., Acta Sci. Math.

(Szeged)

7

_(1935),

95-102.

[3]

A.

_IVI0106,

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

_(1985).

[4]

A.

_IVI0106,

On a _{multiplicative function} connected with the number _of direct _factors

of a _finite Abelian group, (to appear).

[5] E.

_KRÄTZEL,

Lattice Points, Kluwer Academic _Publishers, Dordrecht-

Boston-London, (1988).

[6]

E.

_KRÄTZEL,

On the average numbers of direct factors of a _finite Abelian _group,

Acta Arith. 11

_(1988),

369-379.

[7]

E.

_KRÄTZEL,

Estimates in the General Divisor Problem, Abh. Math. Sem. Univ.

Hamburg 62 _(1992), 191-206.

[8] H.-Q. LIU, On the number _of Abelian groups of a _{given order,} Acta Arith. 56 _(1991),

261-277.

[9]

H. _{MENZER, Exponentialsummen} und verallgemeinerte Teilerprobleme

Disserta-tion, B. Friedrich-Schiller-Universitaet _{Jena, (1991).}

[10]

H. MENZER &#x26; R. _SEIBOLD, On the average number of direct factors of a _finite

Abelian _group., Monatsh. Math. 110 _(1990), 63-72.

[11]

M. VOGTS, Many-dimensional Generalized Divisor Problems, Math. Nachr. 124

(1985),103-121.

Hartmut MENZER

Fakultaet fur Mathematik und Informatik

Friedrich-Schiller-Universitaet Jena D-07740 Jena, Allemagne