ASYMPTOTIC
FORMULAS
FOR
ARITHMETIC
FUNCTIONS
P.
Erdlr(s
April9,
(received
SOME
1958)
Let f(x) be an increasing
function.
Recently
been several
papers
which
proved
that under
fairly
ditions
on f(x) the density
of integers
n for which
is 6/n’
and that (d(n) denotes
the number
of divisors
x
d(n,
n= 1
t
In particular
and the first
[f(nd
note
we are
d(n,[en]
(2)
should
for
d
c
every
a/b
going
to prove
1. The necessary
d we should
have
THEOREM
an irrational
is that
T2x/6.
both of these results
hold if f(x) = x “,
holds if f(x) = [q xl
, a irrational.
In this
for
) = (( 1 + o(l))
‘) there
have
general
con(n, f(n) )= 1
of n )
) = (1 +o(l)
c 7 0
the
oc+
l/(l+~)~
d
be finite
Denote
in positive
e’(n)
O(4 1
the following:
and
sufficient
condition
that
) I12x/6
number
integers
of solutions
a and
It is easy
= fd.
04
of
b.
to see that
for
04 P(e i
din
= (1 + o(l) ) xlogx
t3) 21 = 1 cftn, Ln’l)
Very likely
(3) also holds for l/2 C d C
yet been able to show this.
By more
complicated
can show
Can.
Math.
Bull.
, vol.
1,
no.
3, Sept.
143
1958
1 but I have
arguments
not
I
THEOREM
an irrational
for
2.
d
The necessary
we should
have
dh cn41 1= ( l/2
(4)fl
and
+ o( 1) ) x log
is that for every
E > 0 the number
gers a and b of
and
sufficient
condition
that
x
of solutions
in positive
inte-
of
(6)
~4 <
should
a/b
< o( +
Eb-2/log
b
be finite.
It is easy
to the following:
(l/n)
log
to see that conditions
(5) and (6) are
1 . . . , then
Put d= ao + -- 1
a1t a2+
an3
0, (l/n) a 2n+l
-+
O*
In the present
note we will not prove
proof
is similar
to that of Theorem
1, but
plicated.
Similarly
one could
try
to obtain
Theorem
is rather
equivalent
2 since the
more
com-
an asymptotic
formula
for
X
-2
b(n,
[f(nl-J
1
n=l
for more
general
functions
f(x),
taining
any interesting
results.
Now
of integers
we prove
Theorem
1.
1 c n ( y for which
(n,[n 4-J): 0 (mod
vk
that
is
(n,
ei [n 42
uk+/3
but I have
Denote
k) holds
not
succeeded
by N(y,
l/k)
if and
only
if
if and only
if
, O<fj<l,
) I 0 (mod
k) holds
150
in ob-
the number
n = vk
and
Thus the number
(mod k) equals
by interchanging
(7)r; l
d(n 9 En
I
Since
nd
-
(8)
N
(x/k,
for fixed
irrational
of integers
n( x
satisfying
(n,
N (x/k,
l/k), (since
n - vk implies
the order
of summation
[n 4 ]
l/k)
Ln
(10)
- (1 + 4 1) )
ol
c
Put
a/b
that
4
(14c)b
2
(12)
N
Thus
from
(1 c
o(1)
Inr2,
we have
Thus
(vb,&vb
,
1) we evidently
have
from
(7) and
)~&x/k2
satisfied.
of b for
(8) for
every
= (1+0(l)
Then
which
there
kr2,/6
is a fixed
l/( l+~)~,
Write
n+O(modb),
in
s 2, n= 0 (mod b) . From
it follows
that for fixed
k
Ocn$
-
[noc]Ll/k
x/k2
- x/b.
than
{x/k,
(13Eiq
l/k)
,
Thus
where
in 21,
n * 0 (mod b) and
the equidistribution
of n =L- [ng
the number
of integers
satisfying
is not less
N(x/k,
(x/k2)
(2) is not
large
values
0l.c
= x.
14 ncx,
1
(mod
to infinity.
a( ] )
Assume
now
c > O and arbitrarily
I
is equidistributed
k as x tends
4
(9)znfl d(n,
d 3 ) =zkx
c n 4 ] ) L0
VG x/k).
Thus
(7)
l/k)
-
and (12)
x/b
=
(1 +0(l))
for
every
fixed
((1 + o(l) ) x/k’)
t
k=l
n = vbC - x.
Thus from
- tx/b
= (1 + o( 1) ) v2x/6.
d-j)
s
we have
0 (mod
151
v)
(10)
for
t
and vb&
all
15
v<
x,
x/b.
(1~)~
- x
Hence
(14& 3 r:
d(v)
15~
Now (111,
a necessary
= (1 + o( 1) )(x/t)
log(x/b)
= (1
log(l
‘5%
+ o(l))x
(13) and (14) show that (1)
condition
for the validity
+ c)
does not hold,
of (1).
Thus
(2)
is
To show that (2) is sufficient
we need an upper
estimation
for N (x/k,
l/k) for large
k.
Put x/k - y : it is well known
that
there
exists
an a/b satisfying
Id-a/b}
<
l/(W)
Now we distinguish
for 15 n L y
two
Clearly
(16)
kd.
Thus
where
O<na-
( 17)
z/b
(15)
-
Ltr4-j
=
, b
(
cases.
u/b
2y,
First
+
(a,b)
= 1.
assume
\e\
O/b,
b > k/2.
<
l/2
[no(]<l/kcanonlyholdifu=O,
L
l/k
< (z *1)/k
, or
.
1, . . ..z+
z L
b/k
1
.
The number
of n’s not exceeding
y for which
u has a given
value
is clearly
less than 2y/b + 1, Thus from
(17) and b > k/2
we have
(18)
N (x/k,
l/k) c (b/k
+ 1) (2y/b
+ 1) 2 (3b/k)
Next assume
b< k/2.
If a/bL
ti then
in (16) 8 L 0, thus for u = 0 nP1forusli0(End]
>
1,2b ::,l?
.
Thus a/b > ti , Clearly
if u - 0, that is if n L 0 (mod
(19)
Nx/k,
l/k)
b
If N(x/k,
l/k)
Oc n olLn al<l/k
b 4
but
(20)
this
b/log
-Lb
implies
x +
O< n
b).
d
- [n
Thus
d]c
(4y,b)
N(x/k,
is not
l/k
= 12x/k’
l/k)
in (0,
is only
q
0 since
l/k) and
possible
jx,(bk).
> 0, then (since
all the n < x/k for which
are multiples
of b) we have by (15)
< 1 c min
(k/x,
by (2) that
m.
152
l/k)
L
x-1,2
,
.
fixed
Thus
t
finally
from
(7),
(8))
(18)
and (19)
we have
for
every
X
>
non
d(n,
)L
(1 +o(l))z&c/6
+12x
2
k>t
n-1
hence
(21)
we+wvL$
44%
by (20)
x
d(n,
z
From
[n
*]
&
(1
+0(l)
)srz,/6
.
n-1
(9)
and (21)
we have
that
if (2)
is satisfied,
then
X
z
nll
d(n,
Thus condition
our Theorem.
University
) = (1 +0(l))
Lcl
(2)
is sufficient,
-of British
W2x/6
which
.
completes
the proof
of
Columbia
1) See G .L. Watson,
Canadian
Journal
of Math.
5( 1953), 451-455,
ibid 5( 1953))
456-459
and J. Lambek
and
T. Estermann,
L. Moser,
ibid 7(1955),
155-158.
See also a forthcoming
paper
by P. Erdgs
and G.G.
Lorentz
in Acta Arithmetica.
CORRECTION
In the paper
“On an elementary
problem
by Paul Erdb’s
in Vol. 1, no. 1 of this Bulletin,
should
read
0 L,
u,v
c
f(x)
and
(s&u,
y+v)
153
#
1.
in number
theory”
P. 5 , line 5