81
I ntern. J. Math. & Math. Sci.
Vol 2 (1979) 81-90
GAUSSIAN INTEGERS WITH SMALL PRIME FACTORS
D.G. HAZLL:WOOD
Department of Mathematics
Southwest Texas State University
San Marcos, Texas 78666
U.S.A.
(Recleved April 18, 1978)
ABSTRACT.
exceeding x
Let
2t
G(xt,x)
denote the number of Gausslan integers with norm not
whose Gausslan prime factors have norm not exceeding x
2
Previous
estimates have required restrictions on the parameter t with respect to x.
purpose of this note is to present asymptotic estimates for
G
t
(x x) for all
ranges of the parameter t with respect to x.
KEY WORDS AND PHRASES.
Gausian integr, small prime
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
IOH40.
faors.
Primary 10-02, IOH15, IOH25,
The
82
i.
D.G. HAZLEWOOD
INTRODUCTION.
Let
=
Throughout the discussion
6, e arbitrary positive constants.
the norm of u, and
.
denote a Gaussian integer, p a Gaussian prime, N
the constants implied by the use of
the0-notatlon will
be absolute unless other-
wise indicated.
For real numbers x
=>
1 and t
_>_
0, J. H. Jordan [3] and the author [2] gave
asymptotic estimates for the number of Gausslan integers with norms not exceeding
x
2t
having only Gausslan prime factors with norm not exceeding x
2
However,
Jordan’s estimate fixed the parameter t and our estimate had t bounded with
The purpose of this note is to present an asymptotic estimate for
respect to x.
all ranges of the parameter t with respect to x.
2.
MAIN RESULTS.
THEOREM.
exceeding x
TG(Xt,X)
Let
2t
denote the number of Gausslan integers with norm not
having only Gausslan prime factors with norm not exceeding x
2
Then:
i)
If
t
=<
(log
x)3/5-,
then
ux2t(z(t)
G(xt,x)
+
)e "iogttZ(t)x"
(2.1)
for t outside the interval (I, I+) where Z(t) (the well-known Dickman function)
satisfies the equation
t
with initial condition Z(t)
Z(t)
ll)
If (log
z’ (t) =-z(t-1)
1 for 0 < t < I.
exp{-t(log t + log log t
x)3/5-6 < t
G (xt x)
x
2t
__< x/log
1
+
Further as t
log log
log t
t)
/
(logt t ))
x, then
exp{-t(log t + log log t
I) +
(t
log log
log t
t)}
(2 2)
GAUSSIAN INTEGERS WITH PRIME FACTORS
If x/log x < t <
iii)
TG(xt,x)
If t >
iv)
x2/(elog
Tc(Xt,x)
G(X)
where
x2/(elog
x
2t
83
x), then
exp{-t (log t + log log t
+0(i))}
(2.3)
x), then
exp{C(X)
log t
x
2
+
x
2
2
21gloglgx x +0(l.og x)
(2.4)
denotes the number of Gaussian primes with norm not exceeding
PROOF of the Theorem.
behavior of Z(t)
x
2
Now Case i) follows from Theorem 5 of [2] and the
To derive (2.2) and (2.3) we will first write a lower
G(xt,x)
estimate and then an upper estimate for
that are of the same order.
For the lower estimate we follow the manner of A. S. Fainleib [I] (for
rational integers) and consider the sum
log Na
Nu< x
p[c
2t
(2.5)
=> Np < x
2
We easily see that (2.5) does not exceed
Tc(Xt’X)’
2t log x
and after some routine calculation, (2.5) is at least as large as
2 (I
t- -2u
) x 2t log x
x
u, x)
du
t-I
for 0 <
< I (fixed) and e
e(x) << exp(-a(log x)
315)
for a an absolute positive
constant,
Thus we have
PG (xt’x) >
x-2 G(xu, x)du
O-)
t
t-I
(2.6)
84
D.G. HAZLEWOOD
Now we let
t Z
Zl(t)
be defined by the equation
l(t)
with initial condition
Z l(t)
+ (I-) Z l(t-)
Zl(t)
t for 0 _< t
(I-) Z l(t-1)
: By Lemma
__<
1 of Fainleib [I],
ast/(R)
Z l(t)
where b
b0t
0
and b
+ b I + exp{-t(log
+ log
t
log t -i)
+t logloglog t)
t
It is easy to see that for t
I are real numbers.
I
(l-e)
Z(t)
’t
>=
1
tZ (u)du
1
t-i
Now we let
-2t
K(t,x)
x
G (xt ,x) XZ(t)
where X is a sufficiently small positive real number.
K(t,x)
=> 0,
Then for 0 < t
_< I,
and for t > 1
K(t,x)
=>
jt-
(1-e)t
K(u,x)du.
t-1
Therefore by Lemma 2 of Fainleib [I], K(t,x) > 0 for all t > 0 so that
G(xt,x)
>
X x
2t
Z(t)
which in turn, implies that
xt,x)-
>
x2texp{-t(log
t
+
log log t
I) +
(t
log log
log t
t) +te)},
(2 7)
for t > i which is the lower estimate that we need.
Now we
follow the manner of B. V. Levin and A. S. Fainleib 4] to obtain the
following general results which as a special case of Lemma 2 give the required
upper estimate for
G(xt,x).
GAUSSIAN INTEGERS WITH PRIME FACTORS
LEMMA I.
Let
be a completely multiplicative non-negative function
f
x < 0
satisfying for
where
h(u)
.
< x
N0
< y2
%f(N0 r)
=O((iog
))-i,
NO,
+
R(x2,y 2)
real number, D is an absolute constant,
T is a
and
(u log u)
lR(u,y2)[
-I
-
du <
Further assume for every Gaussian prime 0 and s > -i
for all 6 > 0.
f(Nor)NO -rs
log P(-l,y)
h(min(x2,y2))
+ D +
log(min(x,y))
T
f(N0r)-log
u
y < 0
and
r2f(N0 r)
N0
85
converges.
!
Y
0 <
Then, if
2 1-
8 (’I-) log y +
log
7
(7!$)
+
(y) -< 1- 1/log y,
0( y2
(I-)
+0(
that
R(u,y
l2
+
log y
6
-1
u log u)
du)
2
(2.9)
with
I
P(s,y)
N0<y
2
(I +
I
f(N0 r)
rs)
implies the index is over only those Gaussian primes 0 in the first
where the
quadrant of the complex plane or on the positive real axis.
PROOF.
Let x > y.
Applying Abel’s summation, we find that
%f(N0 r)
r
<-x2 NO
NO
2
NO -< y
rs
log NO
F(y2)
r
y
2s
log y
y2
F(u)d(
22
x
+
2
where F(u)
log u
+ D + h(u).
1
u61og
R(x2,y
+
u
x
2s
2
R(u, y2
s io8 u+l
du
2
s+l
u
log u
log x
2
D. G. HAZLEWOOD
86
Assume s > 0, and letting x tend to
we obtain
2
N0<_y
2
r=l N0
rs
logN0
r
y
2s
f(N0 r) N0
r=l
sufficiently large, we have
Now since
log y
) +
F(u)d (- s
2
u
-rs <
1 for all 0 such that N0 <
"
log P(s,y)
NO
s io
s+l
R(u
log u
u
y2
s
u+l
du.
2
log u
if s is
Af(N0 r)
<=y2
51o
r=l
rs
log NO
r
Hence by the uniqueness of analytic continuation
Y
F(Y2)
log P(-l,y)
y
2(6-1)
log y
12
2
2
F(u)d
u
6-1
+
log u
]R(u,y2)
(6-I)
u
6
lg u+l
du,
2
log u
(2.10
for all 6 > O.
Substituting F
l(u)
+ h(u)
for F(u) where F
T
l(u)
log u
+ D,
we see that the
right hand side of (2.10) is equal to
2
z__
2
(u61g u) -ldu +0(1) +O(y2(1-6)log-2y) +C)(
4
2
(u61og2u) -ldu)
+
2
[R(u,y2) (u61g u)-Idu)
+0(
2
with
2
(u61g u)-ldu
2(I-6)
2(1-6)
+ O( Y’
(1-6) log y 2
(i-6) 21og2y
Y
Therefore substituting these results into (2.10) we get (2.9) to prove
GAUSSIAN INTEGERS WITH PRIME FACTORS
In the following lemma, we shall require that
R(x2,y 2
87
R(x2,y 2)
satisfy the condition
)
og min (x,y
(2.11)
which is generally satisfied when
f(N0
LEMMA 2.
Let
f
r
NO)
=0( IOEr
Np’
log NO
be a completely multipllcatlve non-negative function
satisfying (2.8) where
R(x2,y 2)
satisfies (2.11) and let
Na f(Na)
F(xt’x)
Then for every
F(x
t
x) < x
2t
such that
t
exp{-t(log
t
.e
t
Na < x
2
P a = Np_<x
2
< t <
+ log log
log x
8e
io8) + Isloglogt t) +0(io gt t
+0(log log x) +O(x x
t -(i
+
2t21g2tlog
PROOF.
+
(2.12)
Let 0 < 6 < 1- I/(21og x), then
F(x
t
x)
x2t6
2
Na<
<
p
]a
f(Na)
(Na)
x
2t6
P(I-6, x)
x
2
=>Np-<x
Using Lemma I, then
F(x t x) < x2t6
exp{
2(-)
(1-6) log x +
x
T
Zog
O(
2 (1-6)
x
1-6
21o g2x
+
+0 [R(u’x2)[ (uzg u)-’du) + Zog Cf}
where
Now
Cf
is an absolute constant depending on f.
D. G. HAZLEWOOD
88
(u,x
where
z
2) (u61og
u)
21og x.
Idu
2(I-2)
log x
+ log log
+
x
exp(-6uz
+-[u]+’l du)
u
Further
exp (-6uz
2 1-2
du
uz
+ [u]+i
)--
(.xlog
x
Hence,
T
F(x t,x) -< x 2t6 exp{
2(I-6)
x
(1-6) log x +
-/f) + 0
x
og(
+C(’x2i{
2(1-6)
(I-6)
21o g2x
") +
(i-26)
+ log
Cf}
(2.13)
Now if we let
(log t
21og x
+ log log
t
log
T
+
log log t
log t
in (2.13) we get (2.12) to comlpete the proof of Lemma 2.
For the special case f(Ne)
Ne
-I
we see that
T
8 so that by Lemma 2, if
2
e<t <
1 x
e log x
G(X
then
t
2t
x) <- x exp{-t(log t + log log t-I + log log
t
log
t21g2t
+0 (l---{gt + 0(log io g x) +0 (x21og
x)
t
t)
+
(2.14)
GAUSSIAN INTEGERS WITH PRIME FACTORS
which is the required upper estimate for
G
(xt x)
Combining (2.7) and (2.14), we derive Cases il)
Finally, for Case iv), fix y
manner of Levln and Falnleib
x
t
y2
and
ili)
of the Theorem.
2
1 x
where t >
e log x
[4] by letting
integers with norm not exceeding
89
Fk(Y)
We again follow the
denote the number of Gausslan
whose prime factors are
"aong" the
first k
Gaussian primes, i.e., we can arrange the Gaussian primes lying either in the
first quadrant of the complex plane or on the positive real axis by the relation
i < j if
N0 i
=< N0j.
We see that
()
Fk-
F (Y)
k
_21ogx
=logN0k
where
Ok
denotes the k-th Gaussian prime lying either in the first quadrant of
the complex plane or on the positive real axis.
Now
F0 (y)
4 and
4(I +
Fy)
2loglog2y]
-<
Therefore proceeding by induction on k
4
>_ I,
I
2 log y
log 2
since t >
2
x
e log x
1
we see
that
Fk(Y) __<
k
Where
k
4
i
log
NOv
2k
k’.
k
2
k(log y)k
(2.15)
90
D.G. HAZLEWOOD
Similarily,
Fk(Y >
I
G (x),
Now we let k
G(xt,x)
2k(log y)k
1
4
k
k
(2.16)
then
Fk(y )
=< exp{G(X)
log 2t
x
Fk(Y)
=> exp{G(X)
log 2t
x
2
+
x
2
+
x
log log
2 log x
x+ (lo-g’x
and
G(xt,x)
2
log log x
2 log x
2
+
)}
which implies (2.4) to conclude the proof of the Theorem.
ACKNOWLEDGMENT.
Partially supported by the Southwest Texas State University
Organized Research Program.
REFERENCES
I.
Falnlelb, A.S. An estimate from below for the number of numbers with
small prime factors, Dokl. Akad. Nauk. UzSSR, No. 7 (1967 3-5 (Russian).
2.
Hazlewood, D.G. Sums over Gausslan integers, Trans. Amer. Math. Soc.,
209 (1975 295-309.
3.
Jordan, J.H.
4.
Levin, B.V. and A.S. Fainlelb Application of certain integral equations
to questions of the theory of numbers, Uspehi Mat. Nauk, 22 (1967), No.
(135), 119-197 Russ. Math. Surveys, 2__2 (1967),No. 3, 119-204.
The divisibility of Gaussian integers by large Gaussian
primes, Duke Math J., 3__2 (1965), 503-509.
3