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```ACTA ARITHMETICA
LXXXI.4 (1997)
Normality of numbers generated by
the values of polynomials at primes
by
Yoshinobu Nakai (Kofu) and Iekata Shiokawa (Yokohama)
To the memory of Norikata Nakagoshi
1. Introduction. Let r ≥ 2 be a fixed integer and let θ = 0.a1 a2 . . . be
the r-adic expansion of a real number θ with 0 < θ < 1. Then θ is said to
be normal to base r if, for any block b1 . . . bl ∈ {0, 1, . . . , r − 1}l ,
n−1 N (θ; b1 . . . bl ; n) = r−l + o(1)
as n → ∞, where N (θ, b1 . . . bl ; n) is the number of indices i ≤ n − l + 1
such that ai = b1 , ai+1 = b2 , . . . , ai+l−1 = bl . Let (m)r denote the radic expansion of an integer m ≥ 1. For any infinite sequence {m1 , m2 , . . .}
of positive integers, we consider the number 0.(m1 )r (m2 )r . . . whose r-adic
expansion is obtained by the concatenation of the strings (m1 )r , (m2 )r , . . .
of r-adic digits, which will be written simply as 0.m1 m2 . . . (r).
Copeland and Erdős [1] proved that the number 0.m1 m2 . . . (r) is normal
to base r for any increasing sequence {m1 , m2 , . . .} of positive integers such
that, for every positive % < 1, the number of mi ’s up to x exceeds x% provided
x is sufficiently large. In particular, the normality of the number
0.23571113 . . . (r)
defined by the primes was established. Davenport and Erdős [2] proved that
the number
0.f (1)f (2) . . . f (n) . . . (r)
is normal to base r, where f (x) is any nonconstant polynomial taking positive integral values at all positive integers.
In this paper, we prove the following
Theorem. Let f (x) be as above. Then the number
α(f ) = 0.f (2)f (3)f (5)f (7)f (11)f (13) . . . (r)
1991 Mathematics Subject Classification: 11K, 11L.
[345]
346
Y.-N. Nakai and I. Shiokawa
defined by the values of f (x) at primes is normal to base r. More precisely,
for any block b1 . . . bl ∈ {0, 1, . . . , r − 1}l , we have
1
−1
−l
(1)
n N (α(f ); b1 . . . bl ; n) = r + O
log n
as n → ∞, where the implied constant depends possibly on r, f , and l.
2. Preliminary of the proof of the Theorem. Let α(f ) =
0.a1 a2 . . . an . . . be the r-adic expansion of the number α(f ) given in the
Theorem. Then each an belongs to the corresponding string (f (pν ))r , where
pν is the νth prime and ν = ν(n) is defined by
ν−1
X
ν
X
([logr f (pi )] + 1) < n ≤
([logr f (pi )] + 1).
i=1
i=1
Here [t] denotes the greatest integer not exceeding the real number t. We
put x = x(n) = pν(n) , so that
X
(2)
n=
logr f (p) + O(π(x)) + O(logr f (x))
p≤x
x
dx
+O
=
,
log r
log x
where d ≥ 1 is the degree of the polynomial f (t), p runs through prime
numbers, and π(x) is the number of primes not exceeding x. We used here
the prime number theorem:
x
π(x) = Li x + O
,
(log x)G
where G is a positive constant given arbitrarily and
\
x
Li x =
2
Then we have
N (α(f ); b1 . . . bl ; n) =
X
p≤x
=
X
p≤x
dt
.
log t
N (f (p); b1 . . . bl ) + O(π(x)) + O(logr f (x))
n
N (f (p); b1 . . . bl ) + O
log n
with x = x(n) = pν(n) .
Let j0 be a large constant. Then for each integer j ≥ j0 , there is an
integer nj such that
rj−2 ≤ f (nj ) < rj−1 ≤ f (nj + 1) < rj .
Normality of numbers
347
We note that
nj rj/d
and that nj < n ≤ nj+1 if and only if the r-adic expansion of f (n) is of
length j; namely,
(f (n))r = cj−1 . . . c1 c0 ∈ {0, 1, . . . , r − 1}j ,
(3)
cj−1 6= 0.
For any x > rj0 , we define an integer J = J(x) by
nJ < x ≤ nJ+1 ,
so that
(4)
J = logr f (x) + O(1) log x.
Let n be an integer with nj < n ≤ nj+1 and j0 < j ≤ J, so that (f (n))r
can be written as in (3). We denote by N ∗ (f (n); b1 . . . bl ) the number of
occurrences of the block b1 . . . bl appearing in the string 0| .{z
. . 0} cj−1 . . . c1 c0
of length J. Then we have
J−j
X
X
0≤
N ∗ (f (p); b1 . . . bl ) −
N (f (p); b1 . . . bl )
p≤x
≤
J−1
X
p≤x
(J − j)(π(nj+1 ) − π(nj )) + O(1)
j=j0 +1
≤
J−1
X
π(nj+1 ) + O(1) j=j0 +1
J−1
X
j=1
rj/d
x
J
log x
and so
(5)
N (α(f ); b1 . . . bl ; n) =
X
p≤x
n
N (f (p); b1 . . . bl ) + O
log n
∗
with x = x(n) = pν(n) .
We shall prove in Sections 4 and 5 that
(6)
X
p≤x
x
N (f (p); b1 . . . bl ) = r π(x) logr f (x) + O
log x
∗
−l
which, combined with (5) and (2), yields (1).
348
Y.-N. Nakai and I. Shiokawa
3. Lemmas
Lemma 1 ([9; 4.19]). Let F (x) be a real function, k times differentiable,
and satisfying |F (k) (x)| ≥ λ > 0 throughout the interval [a, b]. Then
b\
e(F
(x))
dx
≤ c(k)λ−1/k .
a
Lemma 2 ([3; p. 66, Theorem 10]). Let
h d
t + α1 td−1 + . . . + αk ,
q
where h, q are coprime integers and αi ’s are real. Suppose that
F (t) =
(log x)σ ≤ q ≤ xd (log x)−σ ,
where σ > 26d (σ0 + 1) with σ0 > 0. Then
X
e(F (p)) ≤ c(d)x(log x)−σ0
p≤x
as x → ∞, where p runs through the primes.
Lemma 3 ([3; p. 2, Lemma 1.3 and p. 5, Lemma 1.6]). Let
F (x) = b0 xd + b1 xd−1 + . . . + bd−1 x + bd
be a polynomial with integral coefficients and let q be a positive integer. Let
D be the greatest common divisor of q, b0 , b1 , . . . , and bd−1 . Then
q X
F (n) 3ω(q/D) 1/d 1−1/d
e
D q
≤d
q
n=1
as q → ∞, where ω(n) is the number of distinct prime divisors of n.
Lemma 4 ([6; Corollary of Lemma]). Let F (x) be a polynomial with real
coefficients with leading term Axd , where A 6= 0 and d ≥ 2. Let a/q be a
rational number with (a, q) = 1 such that |A − a/q| < q −2 . Assume that
(log Q)H ≤ q ≤ Qd /(log Q)H ,
where H > d2 + 2d G with G ≥ 0. Then
X
e(F (n)) Q(log Q)−G .
1≤n≤Q
Lemma 5 ([7; Theorem], cf. [8; Theorem 1]). Let f (t) and b1 . . . bl be as
in Theorem. Then
X
N (f (n); b1 . . . bl ) = r−l y logr f (y) + O(y)
n≤y
as y → ∞, where the implied constant depends possibly on r, f , and l.
Normality of numbers
349
4. Proof of the Theorem. We have to prove the inequality (6). We
write
J
X X
X
f (p)
N ∗ (f (p); b1 . . . bl ) =
I
,
rm
p≤x
where
I(t) =
p≤x m=l



1
if



l
X
bk r−k ≤ t − [t] <
l
X
bk r−k + r−l ,
k=1
k=1
0 otherwise.
There are functions I− (t) and I+ (t) such that I− (t) ≤ I(t) ≤ I+ (t), having
Fourier expansion of the form
∞
X
−l
−1
I± (t) = r ± J +
A± (ν)e(νt)
ν=−∞
ν6=0
with
where e(x) = e2πix
c0 and put
|A± (ν)| min(|ν|−1 , Jν −2 ),
([10; Chap. 2, Lemma 2]). We choose a large constant
(7)
M = [c0 logr J].
Then it follows that
X
(8)
N ∗ (f (p); b1 . . . bl )
p≤x
X
Q
l≤m≤dM
dM <m≤J−M
+
X
J−M <m≤J
X
p≤x
I±
f (p)
rm
P
P
π(x)
(J − dM ) + 2 + 3 + O(π(x)),
rl
where d is the degree of the polynomial f (x),
X
X f (p) P
P
I±
,
1(±) =
1 =
rm
l≤m≤dM p≤x
X
X
X ν
P
P
A± (ν)
e m f (p) ,
2(±) =
2 =
r
p≤x
dM <m≤J−M 1≤|ν|≤J 2
X
X
X
P
P
ν
A± (ν)
e m f (p) .
3(±) =
3 =
r
J−M <m≤J 1≤|ν|≤J 2
p≤x
P
We first estimate 2 . Suppose that dM ≤ m ≤ J − M . Then, writing
the leading coefficient of the polynomial νr−m f (t) as a/q with (a, q) = 1,
=
P
+
X
1
+
350
Y.-N. Nakai and I. Shiokawa
we have
(log x)σ ≤ q ≤ xd (log x)−σ
with a large constant σ, so that by Lemma 2,
X ν
e m f (p) x(log x)−σ0 ,
r
p≤x
where σ0 > 3 is a constant. Therefore we obtain
P
x
2−σ0
(9)
.
2 x(log x)
log x
P
Next we estimate 3 . We appeal to the prime number theorem of the
form referred to in Section 2. Then it follows that
x\ X ν
ν
e m f (p) = e m f (t) dπ(t) + O(1)
r
r
2
p≤x
x
\ ν
dt
x
= e m f (t)
+O
r
log t
(log x)G
2
x
\
dt
x
ν
+O
=
e m f (t)
r
log t
(log x)G
−G
x(log x)
1
sup log x ξ
1
|ν|
log x rm
\
ξ
ν
x
e m f (t) dt + O
r
(log x)G
−G
x(log x)
−1/d
x
+O
,
(log x)G
using the second mean-value theorem and Lemma 1 with |νr−m f (d) (t)| |ν|r−m . Therefore we have
−1/d
X
X
P
1
|ν|
x
−1
|ν|
(10)
+O
3 log x rm
(log x)G
J−M ≤m≤J
1≤|ν|≤J 2
X
X
1
x
1
−m/d
r
+O
log x
(log x)G−2
|ν|1+1/d
2
1≤|ν|≤J
m≤J
x
.
log x
To prove the Theorem, it remains to show that
P
π(x)
x
dM + O
,
(11)
1 =
rl
log x
Normality of numbers
351
since this together with (4), (8), (9), and (10) implies
X
π(x)
N ∗ (f (p); b1 . . . bl ) = l J + O(π(x))
r
p≤x
=
π(x)
x
log
f
(x)
+
O
,
r
rl
log x
which is the inequality (6).
5. Proof of Theorem (continued). We shall prove the inequality (11)
in three steps.
F i r s t s t e p. Suppose that l ≤ m ≤ dM , where M is given by (7)
with (4). We appeal to the prime number theorem for arithmetic progressions of the following form ([4; Sect. 17]): Let π(x; q, a) be the number of
primes p ≤ x in an arithmetic progression p ≡ a (mod q) with (a, q) = 1
and let ϕ(n) be the Euler function. Then
√
1
Li x + O(xe−c log x )
π(x; q, a) =
ϕ(q)
uniformly in 1 ≤ q ≤ (log x)H , where c > 0 is a constant which depends
on a constant H > 0 given arbitrary. (A weaker result O(x(log x)−G ) is
enough for our purpose.) Let B denote the least common multiple of all
denominators of the coefficients, other than the constant term, of f (t). Then
X f (p) X
f (p)
I±
=
I±
+ O(1)
rm
rm
p≤x
p≤x
(p,Br)=1
X
=
I±
a mod Br m
(a,Br)=1
X
=
I±
a mod Br m
(a,Br)=1
f (a)
π(x; Brm , a) + O(1)
rm
f (a)
rm
+ O(1)
π(x)
=
ϕ(Brm )
Hence we have
X
P
(12)
1 Q
l≤m≤dM
X
I±
m
a mod Br
(a,Br)=1
π(x)
ϕ(Brm )
1
x
Li x + O
ϕ(Brm )
(log x)G
f (a)
rm
X
I±
m
a mod Br
(a,Br)=1
+ O rm
f (a)
rm
x
.
(log x)G
+ O M rdM
x
(log x)G
352
Y.-N. Nakai and I. Shiokawa
=
X
l≤m≤dM
=
X
π(x)
ϕ(Brm )
X
µ(b)
l≤m≤dM
b|Br
X
I±
a mod Br m
b|Br
X
I±
a mod Br m
b|a
X
l≤m≤dM
1
Brm
µ(b) + O
b|(a,Br)
X
π(x)
ϕ(Brm )
Br X
= π(x)
µ(b)
ϕ(Br)
f (a)
rm
f (a)
rm
+O
X
I±
1≤n≤Br m /b
x
log x
x
log x
f (bn)
rm
x
+O
,
log x
where µ(n) is the Möbius function. Note that Br = O(1).
S e c o n d s t e p. We shall prove that, for each b | Br,
X
X
1
f (bn)
(13)
I±
Brm
rm
l≤m≤dM
1≤n≤Br m /b
X
X
1
f (bn)
=
I±
+ O(1).
BrM
rm
M
l≤m≤dM
1≤n≤Br
If l ≤ m ≤ M , then we have
X
1
f (bn)
1
I±
=
m
m
Br
r
BrM
m
1≤n≤Br /b
so that
(14)
X
l≤m<M
1
Brm
X
I±
1≤n≤Br m /b
=
f (bn)
rm
/b
X
I±
1≤n≤Br M /b
f (bn)
,
rm
X
l≤m≤M
1
BrM
X
1≤n<Br M /b
I±
f (bn)
.
rm
If d = 1, (14) implies (13). So in what follows we assume d ≥ 2 and M ≤
m ≤ dM . We have
X
f (bn)
I±
rm
1≤n≤Br m /b
m
X
X
Brm 1
r
1 ν
Q
· l +O
+O
f
(bn)
e
b
r
J
|ν| rm
2
m
1≤|ν|≤J
=
m
r
Brm 1
· l +O
b
r
J
1≤n≤Br /b
+ O(rm(1−1/d) J 2/d log J),
Normality of numbers
since, by Lemma 3,
X
1≤n≤Br m /b
Hence we get
X
(15)
M ≤m≤dM
353
1
Brm
ν
e m f (bn) (rm , ν)1/d rm(1−1/d) .
r
X
I±
1≤n≤Br m /b
f (bn)
rm
=
(d − 1)M
+ O(1).
brl
In the rest of this step, we shall prove the inequality
X
X
1
(d − 1)M
f (bn)
(16)
I±
=
+ O(1),
M
m
Br
r
brl
M
M ≤m≤dM
1≤n≤Br
/b
which together with (15) and (14) yields (13).
P r o o f o f (16). It is easily seen that
X
X
1
f (bn)
(17)
I±
BrM
rm
M ≤m≤dM
1≤n≤Br M /b
X
X
1
1
1
Q
+O
M
l
Br
r
J
M
M ≤m≤dM 1≤n≤Br
/b
X
+
1≤|ν|≤J
=
(d − 1)M
+ O(1)
brl
X
1
1
+O
·
M
|ν|
Br
2
1≤|ν|≤J
ν
A± (ν)e m f (bn)
r
2
X
M ≤m≤dM
X
1≤n≤Br M /b
ν
e m f (bn) .
r
We estimate the last sum. Let H be a large constant. For any ν, m, b, we
can choose, by Dirichlet’s theorem, coprime integers a and q = q(ν, m, b)
such that
1 ≤ q ≤ Qd /(log Q)H ,
and
ν d
b −
rm
a (log Q)H
<
q
qQd
Q = BrM /b
(≤ 1/q 2 ).
If
(log Q)H ≤ q ≤ Qd /(log Q)H ,
354
Y.-N. Nakai and I. Shiokawa
then by Lemma 4,
X
1≤n≤Br M /b
ν
rM
Q
e m f (bn) .
r
(log Q)G
(log J)2
Hence the contribution of these sums in the last term in (17) is
1
rM
(d
−
1)M
log
J
·
= O(1).
BrM
(log J)2
Otherwise, we have
1 ≤ q ≤ (log Q)H
( M H ).
In particular, (ν/rm )bd 6= a/q, since m ≥ M . Hence
H
ν d a
1
M ,
≤
b
−
rm
qrm
q
qrdM
so that
(dM ≥) m ≥ dM − H1 log M,
with a large constant H1 . From this it follows that
d ν
ν
· m f (bt) m td−1 J 2 r−M +H1 log M = o(1)
dt r
r
throughout the interval [1, BrM /b]. Thus by a van der Corput’s lemma ([9;
Lemma 4.8]) we have
X
1≤n≤Br M /b
ν
e m f (bn) =
r
\
Br M /b
1
ν
e m f (bt) dt + O(1)
r
−1/d
ν (d) −1/d
|ν|
m f (t)
,
+ O(1) r
rm
using again Lemma 1. Hence the contribution of these sums to the last term
in (17) is
1
BrM
X
X
M ≤m≤dM 1≤|ν|≤J 2
−1/d
1 |ν|
= O(1).
|ν| rm
Combining these results, we obtain (16).
Normality of numbers
355
T h i r d s t e p. It follows from (12) with (13) that
X
X
P
Br X
1
f (bn)
I±
µ(b) M
1 Q π(x)
ϕ(Br)
Br
rm
M
l≤m≤dM 1≤n≤Br /b
b|Br
x
+O
log x
X
X
Br X
1
f (bn)
Q π(x)
I
µ(b) M
ϕ(Br)
Br
rm
M
l≤m≤dM
b|Br
1≤n≤Br /b
x
+O
.
log x
We put, in Lemma 5, y = BrM /b, so that logr f (by) = dM + O(1). Then
we have
X
X
X
f (bn)
I
=
N (f (bn); b1 . . . bl ) + O(rM )
m
r
M
l≤m≤dM 1≤n≤Br
/b
n≤y
= r−l y logr f (by) + O(rM )
= r−l
BrM
dM + O(rM ).
b
Therefore we obtain
P
1 R
Br X µ(b) dM
x
· l π(x) + O
ϕ(Br)
b
r
log x
b|Br
x
−l
= r dM π(x) + O
,
log x
which is (11). The proof of the Theorem is now complete.
References
[1]
[2]
[3]
[4]
[5]
[6]
A. H. C o p e l a n d and P. E r d ő s, Notes on normal numbers, Bull. Amer. Math. Soc.
52 (1946), 857–860.
H. D a v e n p o r t and P. E r d ő s, Note on normal decimals, Canad. J. Math. 4 (1952),
58–63.
L.-K. H u a, Additive Theory of Prime Numbers, Transl. Math. Monograph 13, Amer.
Math. Soc., Providence, RI, 1965.
M. N. H u x l e y, The Distribution of Prime Numbers, Oxford Math. Monograph,
Oxford Univ. Press, 1972.
Y.-N. N a k a i and I. S h i o k a w a, A class of normal numbers, Japan. J. Math. 16
(1990), 17–29.
—, —, A class of normal numbers II , in: Number Theory and Cryptography,
J. H. Loxton (ed.), London Math. Soc. Lecture Note Ser. 154, Cambridge Univ.
Press, 1990, 204–210.
356
[7]
[8]
[9]
[10]
Y.-N. Nakai and I. Shiokawa
Y.-N. N a k a i and I. S h i o k a w a, Discrepancy estimates for a class of normal numbers, Acta Arith. 62 (1992), 271–284.
J. S c h i f f e r, Discrepancy of normal numbers, ibid. 47 (1986), 175–186.
E. C. T i t c h m a r s h, The Theory of the Riemann Zeta-Function, 2nd ed., revised
by D. R. Heath-Brown, Oxford Univ. Press, 1986.
I. M. V i n o g r a d o v, The Method of Trigonometrical Sums in Number Theory,
Nauka, 1971 (in Russian).
Department of Mathematics
Faculty of Education
Yamanashi University
Kofu, 400 Japan
E-mail: nakai@grape.kkb.yamanashi.ac.jp
Department of Mathematics
Keio University
Hiyoshi, Yokohama, 223 Japan
E-mail: shiokawa@math.keio.ac.jp