ON SOME
APPLICATIONS
OF
PROBABILITY
TO
ANALYSIS
AND NUMBER THEORY
P
. ERDÖS
It would be quite impossible to give a survey of these subjects in a
short article or lecture, and I will only succeed by making some arbitrary
restrictions on the topics with which I will deal
. First of all, I will restrict
myself to problems and results on which I worked, and secondly, I will
not discuss subjects which have been discussed in recently appeared
review articles
[1]
.
Probabilistic methods have been used in analysis for several decades
;
it suffices to name Paley, Wiener, Kolmogoroff, Zygmund, Salem,
Steinhaus, Kac, Dvoretzky, Kahane, and many others
.
I will restrict
myself to some questions my collaborators and I worked on for several
ar
years
.
Hardy was the first to give an example of a power series
E a,,,
z"k
k=1
cA
which converges uniformly in z a < 1 but for which
a
k
= oo
.
Piranian
k=1
asked me for what sequences of integers
n
1
< n
2
<
. .
.
does there exist a
power series
a
k
zn
k
which converges uniformly in
I
z < 1 but for which
k=1
z
Z I a
k
= 00
.
I proved [2] by probabilistic methods that if the sequence
k=1
{n
1
.}
satisfies
lim inf
(n
;n
1
)
1
i
(
i

0 =
1
where
ji>oo
then if
Y
a,
.
z111
converges for
k1
z
(1)
then such a power series exists
. Zygmund [3] proved that if
n
k+1/n
k
> c > 1
1, Z
I
a, < oo
.
Thus (1) is certainly
k=1
not far from being best possible, and it is quite likely that it is, in fact,
best possible
;
in fact, Zygmund's theorem may remain true for every
sequence which does not satisfy
(1), in other words, for every sequence
{n
k}
for which there exists an absolute constant c so that for every
i <
j
n
;n
ti
>
(
1+c)'

ti .
(2)
Curiously enough, (1) occurred in a seemingly different context
.
Gaier and MeyerKönig [4] call the radius defined by
z = re
2
O,
0 <
it <
1,
U)
singular for
f (z) = Z
a
k
z
n
if
f (z) is
unbounded in every sector
I z I
< 1,
71=1
Co
~ E
< arg
z <
0f
E
where
E >
0
.
They showed that if
f (z) _
a,,
znk
k=1
and
n
k+1/n
k
> c > 1,
and if
f (z) is
unbounded in (
z ~ < 1, then every radius
Received 1 May, 1963
.
[JOURNAL LONDON MATH
. Soc
., 39 (1964), 692696]
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APPLICATIONS OF PROBABILITY
TO ANALYSIS AND NUMBER THEORY
693
is a singular radius
. Rényi and I [5] showed by probabilistic methods
W
that if
{n,
;}
satisfies (1) then there exists a power series
f (z)
= E
a
k
znk
k=1
m
for which
a
k
> 0,
a
k
=
oo, thus the positive real axis is a singular
k=7
radius but no other radius is singular
. In fact,
f (z) is
bounded in
I
z < 1
if a region
I
z1 < e is excluded (for every e > 0) . It again seems quite
possible that our theorem is best possible
;
in fact, perhaps if
{n
k
}
satisfies
(2)
then the theorem of Gaier and MeyerKöng remains true, but we
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 Winter '11
 PhanThuongCang

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