On q-additive and q-multiplicative
functions
By
I.Kátai
Hochiminh City
December, 2010
I. Denitions
q ≥ 2; Aq = {0, 1, . . . , q − 1} =set
n=
∞
X
εj (n) q j ,
j=0
Aq = set of q -additive
f : N0 → R, f ∈ Aq if
of digits
εj (n) ∈ Aq
q − ary
expansion
functions
f (0) = 0,
f (n) =
∞
X
j=0
j
f εj (n) q .
Examples:
1.)
2.)
3.)
∈ Aq
f (n) = cn
α (n) =
∞
P
j=0
εj (n)
n
∈ Aq
βl (n) = # j|εj (n) = l
o
∈ Aq
(l = 1, . . . , q − 1)
Mq = set of q−multiplicative
g : N0 → C, g ∈ Mq if
functions
g (0) = 1,
g (n) =
Examples:
∞
X
j
g εj (n) q .
j=0
1.)
f ∈ Aq , z ∈ C, z 6= 0,
g (n) = z f (n), g ∈ Mq
2.)
g (n) = z n ∈ Mq
3.)
g (n) = wn (x) = the nth
Walsh function ∈ M2
A. O. Gelfond: 1968; H. Delange 1972
Theorem (Delange). Let g ∈ Mq , |g (n) | ≤ 1
(n ∈ N 0 ).
Then
I.

N
Yx 1  X 1 X

j
g cq  + ox (1) ,
m (x) :=
g (n) =

x n≤x
j=0 q c∈Aq

"
#
log x
.
Nx :=
log q
Thus
(1)
always exists.
lim |m (x) | = ξ
x→∞
II.
ξ=0
(2)
or
i either
g cq j
X
=0
for some j ∈ N0,
c∈Aq
∞ X
X
(3)
Re 1 − g cq j
= ∞.
j=0 c∈Aq
III.
lim m (x) = m
(4)
exists and m 6= 0 i
∞ X X
1 − g cq j
j=0 c∈Aq
and (2) does not hold.
is convergent.
Consequence:
i
(5)
f ∈ Aq
has a limit distribution, i.e.
1
lim # {n < x|f (n) < y} = F (y )
x→∞ x
∞ X
X
j=0 a∈Aq
are convergent.
f aq j ,
∞ X
X
j=0 a∈Aq
a.a.y
2
j
f aq
II. On the mean value of t (n) = g1 (a1n) . . . gk (akn)
a1 < . . . < ak (< q ) , ai, aj = 1, i 6= j,
g1, . . . , gk ∈ Mq ;
|gj (n) | = 1 (j = 1, . . . , k) , n ∈ N0.
j
tj (n) := t nq .
t (n) = g1 (a1n) . . . gk (ak n) ;
Mj
qN
=
X
mj
qN
t nq j
n<q N
=
X
tj (n) ;
n<q N
−N
N
:= q
Mj q
,
αj := lim inf |mj (N ) |,
N →∞
(ai, q ) = 1
βj := lim sup |mj (N ) |
N →∞
Theorem 1 (Indlekofer-Kátai: Acta Math. Hung. 2001).
If βj > 0 for some j , then αl = βl (∀l) ; αl → 1 (l → ∞),
and there exist suitable real numbers γ1, . . . , γk and some j0 ≥
0 (j0 ∈ N0) such that
(6)
q j0 (γ1a1 + . . . + γk ak ) ≡ 0 (mod 1),
and in the notation
e (x) := e2πix
hj (n) := e −γj n gj (n) ,
we have
(7)
k X
∞ X
X
Re 1 − hl
cq j
l=1 j=0 c∈Aq
If (6), (7) hold, then βj > 0 for some j .
< ∞.
Let f1, . . . , fk ∈ Aq ,
a1, . . . , ak
as above.
l (n) := f1 (a1n) + . . . + fk (ak n) .
Theorem 2. Let R ≥ 3. Then
max
where
max |fj (nq ) − nfj (q ) | ≤ K max |l (n) |,
R−3
R
j=1,...,k n≤q
K is
a1, . . . , ak .
n<q
an explicit constant, which depends only on
Denition. A sequence {bn}n∈N0
function d (x) such that
bn ∈ R
q,
is tight, if there is a
o
1 n
c (K ) := lim sup # n ≤ x|bn − d (x) | > K ,
x
x
c (K ) → 0 (K → ∞ )
Theorem 3. Assume that l (n) is a tight sequence. Then
there exist γ1, . . . , γk ∈ R such that
(8)
γ1a1 + . . . + γk ak = 0
and for qt (n) := ft (n) − γtn
(9)
∞ X
X
(t = 1, . . . , k)
2
j
qt cq < ∞
(t = 1, . . . , k) .
j=0 c∈Aq
If (8), (9) hold, then
1
lim {n < x|l (n) − d (x) < y} = F (y ) a.a.y
(10)
x
exists, F is a distribution function,
(11)
d (x) :=
k
X
X
j
qt cq .
t=1 cq j <x
Finally, l has a limit distribution, i (11) is convergent.
Let
1
n
o
u+1
πu (c1, . . . , ck ) = u+1 # n < q
|εu aj n = cj , j = 1, . . . , k ,
q
1 X
fl (cq u) ,
µl (u) :=
q c∈A
q
p (u) := µ1 (u) + . . . + µk (u) ,
τ (u) =
X
h
πu (c1, . . . , ck ) f1 (c1q u) + . . . + fk (ck q u)
c1 ,...,ck ∈Aq
− p (u)
2 =
σN
NX
−1
u=0
τ (u) .
i2
,
Theorem 4. Let a1, . . . , ak as earlier, l (n) := f1 (a1n) + . . . +
fk (ak n).
Assume that
max max
Then
l=1,...,k c∈Aq
1
x→∞ x
lim
M
|fl cq
|
σM
→0
(M → ∞ ) .
)
l (n) − p (N )
x
n < x
< y = Φ (y ) ,
σNx
"
#
(
log x
Nx =
log q
These theorems are proved in Acta Math. Hung. 2001, 2002.
III. The analogues of the above problems for
some subsets of integers
Theorem 5. Let f ∈ Aq . Then
on the set of primes (= P ), i.e.
f
has a limit distribution
1
# {p ≤ x|p ∈ P, f (p) < y} = F (y )
lim
x→∞ π (x)
i (12), (13) are convergent.
X X
(12)
f aq j
j a∈Aq
(13)
The main ingredient is
X X
j a∈Aq
2
j
f aq .
a.a.y
F
Lemma 1. Let f ∈ Aq ,
1
m∗0 =
ϕ (q )
1
σ0∗2 =
ϕ (q )
q−1
X
q−1
X
q−1
f (a) ;
a=1
(a,q)=1
2
∗
f (a) − m0 ;
a=1
(a,q)=1
∗ = m∗ + m + . . . + m
MN
1
N −1 ;
0
1 X j
f aq ,
mj =
q a=0
2
1 X j
2
f aq − mj ,
σj =
q a∈A
q
∗2 = σ ∗2 + σ 2 + . . . + σ 2
DN
0
1
N −1 .
Then
X
p<q N
cq
2
N
∗2 ,
DN
(f (p) − MN ) ≤ cq π q
is a suitable constant.
Let B ⊆ N0,
B (x) = # {b ≤ x|b ∈ B} , q N ≤ x < q N +1
(1 ≤)l1 < . . . < lh (< N ) ,
b1, . . . , bh ∈ Aq
l = (l1, . . . , lh) , b = (b1, . . . , bh)
l
b
Conjecture 1.
AB
(3.3)B
!
n
o
x
:= # n ≤ x|n ∈ B, εlj (n) = bj , j = 1, . . . , h .
∃δ > 0
constant, such that
!
l
q hA
B xb
− 1 → 0
sup sup B(x)
h<δN l1 ,...,lh b1 ,...,bh for B = P , and some other arithmetically interesting sequences.
√
Remarks. 1. (3.3)P in weaken form with h < N has been
proved by Kátai in Acta Math. Hung. 1980. Some oversights. Corrected version: G. Harman-I. Kátai in Acta
Arith. 2008.
2. Assume that q > 2, q prime. Then (3.3)P holds with
δ = 1/3, if lh ≤ N/3 (Barban/Linnik/Tsudakov), and if
7 N ≤ l (Prime number theorem for short intervals).
1
12
Theorem 6 (N.
L. Bassily - I. Kátai). Let f ∈ Aq ,
sup |f bq j | < ∞.
j≥0, b∈Aq
Let
Let
h
i
log x
P (x) ∈ Z[x], r = deg P, N = log q , P (x) → ∞
M ( x) =
N
X
mk ,
D 2 ( x) =
k=0
Assume that D(x)(log x)−1/3 → ∞
N
X
as x → ∞.
σk2.
k=0
(x → ∞).
Then
)
r
1
f (P (n)) − M (x )
# n < x
< y → Φ (y )
r
x
D (x )
(
)
r
1
f (P (p)) − M (x )
# p < x
< y → φ (y ) .
r
π ( x)
D (x )
(
The proof is based upon a theorem of I. M. Vinogradov and
a theorem of L. K. Hua.
Lemma 2. Let λ > 0 be arbitrary, h be xed, b1, . . . , bh ∈ Aq ,
N 1/3 ≤ l1 < . . . < lh ≤ rN − N 1/3.
Let
n
o
n
o
Σ1 := # n ≤ x|alj (P (n)) = bj , j = 1, . . . , h ,
Then
Σ2 := # p ≤ x|alj (P (p)) = bj , j = 1, . . . , h .
!
x
x
Σ1 = h + O
;
λ
q
(log x)
!
π ( x)
x
Σ2 =
+O
.
λ
h
q
(log x)
Let
ak (x) :=
1 X
x n≤x
!k
r
f (P (n)) − M (x )
,
D ( xr )
1 X
bk (x) :=
π (x) p≤x
1
ck (x) := r
x n≤xr
X
From Lemma 2
⇒
!k
r
f (P (p)) − M (x )
,
r
D (x )
!k
r
f (n) − M (x )
.
r
D (x )
ak (x) − ck (x) → 0 (x → ∞)
bk (x) − ck (x) → 0 (x → ∞) .
R k
We know: lim ck (x) = x dΦ.
From Frechet-Shohat theorem ⇒ Theorem 6.
How to prove Lemma 2?
g ( x) =
if
if

1
0
x ∈ [0, 1q ]
x ∈ [ 1q , 1]
periodic
b
g b ( x) = g x − q .
Then
Σ1 =
h
X Y
gb j
n≤x j=1
P
Fourier expansion of g (x) ∼
P (n)
q lj +1
!
.
cne (nx),
∆
1
(1)
g
g (x + u) du;
(x) :=
2∆ −∆
Z
∆
1
(2)
g
g (1) (x + u) du,
(x) :=
2∆ −∆
Z
mod 1
g (2) (x) =
X
1
dn = O
n3
dne (nx) ,
trigonometric sums P P 2 j
j
Theorem 7. Let f ∈ A2, f 2 convergent, f 2 < ∞.
Let ξν be independent r.v., P (ξν = 0) = 1−η, P (ξν = f (2ν )) =
η.
Let
Σ1 ∼
θη =
Let 0 < δ < 1/2.
Then
∞
X
ξν ,
Fη (y ) = P (θη < y ) .
ν−1
n
o
1
N
lim
max N # n < 2 , α (n) = k, f (n) < y − F k (y ) = 0
N →∞ k ∈[δ,1−δ] N
N
k
for all continuity points of F 1 (y).
2
Theorem 8. Let
f ∈ A 2 , f 2j
AN :=
Let
= O (1),
NX
−1
j
f 2 .
j=0
j
hN ∈ A2, hN 2 := f 2j − ANN
2 η := 1 − η η
σN
( )
(
)
. Let
NX
−1
2
hN 2j .
j=0
Assume that
Then
2
σN
1 →∞
2
(N → ∞ ).
Let 0 < δ < 12 .
(
)
kAN
1
f (n) − N
N
lim
sup sup N # n < 2 , α (n) = k,
<y
k
N →∞ k ∈[δ,1−δ] y∈R
σN N
k
N
− Φ (y ) = 0.
Theorem 9. Let f ∈
(0, 1) and a sequence
ξ (N → ∞) such that
1
A2. Assume that
kN (N = 1, 2, . . .)
n
there is some ξ ∈
such that kN /N →
# n < 2N |α (n) = kN , f (n) < y
N
kN
o
→ G (y )
a.a.y, where G is a distribution function. Then G (y) = Fξ (y)
dened in Theorem 7, furthermore
X
j
f 2 ,
X
2
f 2j
are convergent.
Theorem 7, 8, 9 are joint results with M. V. Subbarao.
IV. Distribution of q-additive functions on the
set of integers having k prime factors
Let ω (n) be the number of distinct prime divisors of n,
{n|ω (n) = k} , πk (x) = # {n ≤ x, n ∈ Pk }.
It is known that
1
π k ( x) = 1 + O
log log x
x (log log x)k−1
log x
(k − 1)!
Pk :=
uniformly as 1 ≤ k ≤ B log log x, where B is an arbitrary constant. This is a known result of Sathe and A. Selberg.
Theorem 10 (L. Germán-I. Kátai). Let
O (1).
j
f ∈ Aq , sup |f bq | =
j∈N0
b∈Aq
Let Jx = [1, δ (x) log log x], where δ (x) → 0. Then
(
)
1
f (n) − E (x)
sup sup # n ≤ x, n ∈ Pk |
< y − Φ (y ) → 0
D (x)
k∈Jx y∈R πk (x)
(x → ∞ )
where
E (x) :=
N
X
µj ,
D 2 ( x) =
N
X
j=0
j=0
1 X j
µj =
f bq ;
q b∈A
q
"
σj2,
N = Nx =
log x
log q
2
1 X j
2
σj =
f bq − µj .
q b∈A
q
#
V. Number systems over the Gaussian integers
Z[i] = Gaussian integers
θ = a + bi, |θ|2 ≥ 2.
A = {0, a1, . . . , at−1} (⊆ Z[i]), complete residue system mod θ.
J : Z[i] → Z[i].
If α ∈ Z[i], then ∃ unique b0 ∈ A, and α1 ∈ Z[i], such that
α = b0 + θα1. We write J (α) = α1.
Iterating
(14)
Let
α (= α0) , α1 = J (α0) , . . . , αk+1 = J (αk ) , . . . .
L :=
1
max |a|.
|θ| − 1 a∈Aq
Lemma 3.
a) If |α| > L, then |α1| < |α|,
b) If |α| ≤ L, then |α1| ≤ L.
Consequence: (14) is ultimately periodic for every α ∈ Z[i].
set of periodic elements. π ∈ P ∗, if J l (π) = π.
We have:
(15)
α = b0 + b1θ + . . . + bk−1θk−1 + θk αk .
Let k be the smallest integer for which αk ∈ P ∗.
P∗ =
We say that (θ, A) is a number system, if the only periodic
element is 0, i.e. if P ∗ = {0}.
There are a plenty of number systems.
Let (θ, A) be a number system. A function
additive (with respect to (θ, A)) if
f (α) = f (b0) + f (b1θ) + . . . + f bk−1
θk−1
f : Z[i] → R
is
( αk = 0 ) .
The analogues of the theorem of Delange are proved in I. K.-P.
Liardet in Acta Arithmetica.