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AN ASYM PTOT IC FORMULA CONC ERNING

A GENERALIZED EULER FUN CTION

L . T o t h a n d J . S a n d o r

N. Golescu Nr. 5, 3900 Satu-Mare and 4136 Forteni 79, Jud. Harghita, Romania

(Submitted April 1987)

1. Introduction

Harlan Stevens [8] introduced the following generalization of the Euler (p-

f unction. Let

F - {f\(x)

9

. . . , ~f

k

(x)}

,

k >

1, be a set of polynomials with

integral coefficients and let A represent the set of all ordered fc-tuples of

integers (a-,,

. ..,

a^) such that 0 < a-,,

. ..,

a^

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AN ASYMPTOTIC FORMULA CONCERNINGA GENERALIZED EULER FUNCTION

1, for all i and j.

J

^

r

J ^

< < :

Now,

f o r

n =

H-

1

p

ej

, \ ]i

(n)

Q

F

(n) \

= 0 if j

e x i s t s s u c h t h a t

^ > 2;

o t h e r -

w i s e ,

| y ( n ) f l

F

( n ) |

= ( - l )

r

- 0 O h j . . . ^ j )

J

=

Hence,

\\i

(n) tt

F

(n) \ < A^

(n)

for all

n,

whereA =

M

k

>

1.

Ontheother hand,one has

2

w n )

=2

r

< fi ^ +1)=d n),

,7=1

s o

oa(n) 0, we

obtain

|u(n)ft

F

(n)|= 0(n

E

)

,

as

desired.

Lemma

3:

Theseries

\i(n)Q

F

(n)

=i

n

s+l

n .-^i . w

is absolutely convergent

for s > 0, and its sum is

given

by

Ms)

p.

where

N^

denotes

the

number

of

incongruent solutions

of f^ (x) = 0 (mod p).

Proof: Theabsolute convergence followsbyLemma2:

|u(n)ft

F

(n)/n

s+1

|

0 is a

constant

and e > 0 is

such that

s -

e > 0.

Note that

the

gen-

eral termismultiplicative in n, so theseriescan beexpanded intoaninfi-

nite Euler-type product

[3,17.4]:

u(n)fi

F

(w)

/ y(p^)^

F

(p

)\ /

M P ) \ _ ,

L,

= ||

I

2^ r-

1

=

11I 1

~i

J

~

A

F-

From hereon, weshalluse thefollowing well-known estimates.

Lemma

4:

n

s

=

^

+

0(x

s

), s >

1; (4)

n< cc

S + 1 '

1989]

177

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AN ASYMPTOTIC FORMULA CONCERNING

A

GENERALIZED EULER FUNCTION

E A =

0 x

l

-

s

),

0

=

IT II (l i)*II (l~

\ ) + 0(x

l +

) for all e > 0. (10)

For

t =

2,

2(n) =

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AN ASYMPTOTIC FORMULA CONCERNING A GENERALIZED EULER FUNCTION

There are

such pairs and the property (/(a), b) - 1 is true for B(n) pairs of them, where

Bin) =