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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) =