MANFRED G. MADRITSCH
Abstract. The present paper deals with functions acting on the digits of an q-ary expansion. In particular let n be a positive integer, then we call
n = ` X r=0
dr(n)qr with dr(n) ∈ {0, . . . , q − 1}
its q-ary expansion. We call a function f strictly q-additive, if for a given value, it acts only on the digits of its representation, i.e.,
f (n) = ` X r=0
f (dr(n)) . The goal is to prove the following
X n≤N
f (bp(n)c) = µfN logq(p(N )) + N Fq,k`logq(p(N ))´ + O `N1−ε´ ,
where p(x) = α0xβ0+ · · · + αdxβd with α0, β0, α1, β1, . . . , αd, βd∈ R and at least one βi6∈ Z. This result is motivated by results of Nakai and Shiokawa and M. Peter.
1. Introduction
Let q ≥ 2 be an integer. Then we can represent every positive integer n in a unique way as follows n = ` X r=0 dr(n)qr with dr(n) ∈ {0, . . . , q − 1}. (1.1)
We call this the q-ary representation of n with q the base and {0, . . . , q − 1} the set of digits. If a function f acts only on the digits of a representation, i.e.,
f (n) =
`
X
r=0
f (dr(n)qr) ,
where n is as in (1.1), then we call it q-additive. If this action of f is independent of the position of the digit, i.e.,
f (n) =
`
X
r=0
f (dr(n)) ,
then we call f strictly q-additive. In the following we will concentrate on these type of functions. A simple example of a strictly q-additive function is the sum-of-digits function sq which is
defined by
sq(n) =
X
r≥0
dr(n),
where n is again as in (1.1). Strictly q-additive functions have been investigated from several points of view. In the present paper we want to concentrate on arithmetical properties and, in particular, on the summatory function of f on pseudo polynomial values.
Date: February 16, 2010.
2000 Mathematics Subject Classification. 11N37 (11A63). Key words and phrases. additive function, pseudo polynomial.
Before we present the result we want to give an overview on what is known for the summatory function in connection with the sum-of-digits function. One of the first results in that direction is due to Delange [1] who was able to show
X
n≤N
sq(n) =
q − 1
2 N logqN + N F logqN ,
where logq is the logarithm to base q and F is a 1-periodic, continuous and nowhere differentiable
function. Remarkable about this result is the lack of an error term.
The moments of the sum-of-digits function were considered by Kirschenhofer [7] and by Grabner et al. [6]. For different methods originating from analytic number theory such as Mellin’s formula elegant proofs have been shown by Flajolet et al. [2] and Grabner and Hwang [5]. Generaliza-tions to number systems in number fields were done by Thuswaldner [14] and Gittenberger and Thuswaldner [4].
Apart from the sequence of the positive integers others like the primes or the integer values of polynomials are of interest. For the case of primes the summatory function of the sum-of-digits function was investigated by Shiokawa [13]. Mauduit and Rivat considered the distribution in residue classes and the uniform distribution modulo 1 of (αsq(bncc))n≥1(with α an irrational and
b·c the floor function) with 1 ≤ c ≤ 4
3 in [10] and with c = 2 in [9]. The distribution for the case
c = 1 goes back to Gelfond [3].
In order to construct a normal number one is also interested in the distribution of the digits in an expansion. A normal number x ∈ R is, informally speaking, a number in whose digital expansion every block of length k occurs asymptotically equally often. In a paper by Nakai and Shiokawa [11] they constructed such a normal number by concatenating the integer part of a pseudo-polynomial sequence, i.e. a sequence (bp(n)c)n≥1where
p(x) = α0xβ0+ α1xβ1+ · · · + αdxβd
(1.2)
with α0, β0, . . . , αd, βd∈ R, α0> 0, β0> β1> · · · > βd≥ 1 and at least one βi6∈ Z. They gained
the following result as a corollary.
Theorem ([11, Corollary 2]). Let p be as in (1.2). Then X
n≤N
sq(bp(n)c) =
q − 1
2 N logqp(N ) + O(N ) where logq denotes the logarithm to base q.
Now we draw our attention towards a resent result by Peter [12] who considered polynomial sequences of the form αnkn≥1 where α, is an irrational of finite type. In particular he could show the following.
Theorem ([12, Theorem]). Let q, k ∈ N \ 1, and α = 1 or α > 0 an irrational of finite type. There are c ∈ R and ε > 0 such that
X n≤N sq(αnk) = q − 1 2 N logq(αN k) + cN + N F log q(αN k) + O(N1−ε),
with N ≥ 1 and logq the logarithm to base q.
Our aim is now to combine the last two results and to generalize them to arbitrary q-additive functions instead of the sum-of-digits function.
In order to state the result in a more convenient way we need some definitions. For x ∈ R we denote by bxc the floor function and by {x} := x − bxc the fractional part of x, respectively. Let ψ be the centralized fraction function defined by
ψ(x) = x − bxc −1
2 = {x} − 1 2. (1.3)
Moreover, we define µf := q−1 X a=0 f (a) Jf,β(x) := Z x 0 q−1 X a=0 f (a) ψ t −a + 1 q − ψ t −a q tβ11dt, x ≥ 0, q, k ∈ N, and Ff,β(t) := µf(1 − {t}) + 1 βq (1−{t})/βX n≥0 q−nβJq,β(qn−1+{t}), t ∈ R. (1.4)
Now we are able to state our result.
Theorem 1.1. Let q ∈ N \ {1} and f be a strictly q-additive function with f (0) = 0. If p is a pseudo polynomial as defined in (1.2), then there exists ε > 0 such that
X
n≤N
f (bp(n)c) = µfN logq(p(N )) + N Ff,β0 logq(p(N )) + O N1−ε .
(1.5)
Remark 1.2. We can show a similar result if f is q-additive but not necessarily strictly q-additive. In this more general setting one has to keep track of the position of a digit in the q-ary expansion throughout the whole proof. This leads to a more delicate periodic function Fq,β0 which then
depends on the integer value of logq(p(N )) (i.e., the position) also.
The idea of proof is to consider each digit of an expansion according to its position within its expansion. Since we need a bound for the maximal length of an expansion we define
R(N ) := logq(p(N )). (1.6)
Then we will distinguish two ranges for the position which are treated in Sections 2 and 3 separately. In particular, we consider sums of the form
Ur,a(N ) := X N <n≤2N ψ p(n) qr+1 − a q . (1.7)
where a ∈ {0, . . . , q − 1} (the digit) and r (the position) lies in 1 ≤ qr+1≤ Nβ0−1 and
(1.8)
Nβ0−1≤ qr+1≤ p(2N ),
(1.9)
respectively. Finally in Section 4 we apply those estimates in order prove Theorem 1.1. 2. Exponential sums
This section focuses on the estimation of (1.7) for r such that (1.8) holds. In particular we will show the following.
Proposition 2.1. Let N be positive and r be an integer such that (1.8) holds, then there exists ε > 0 such that
Ur,a(N ) N1−ε
(2.1)
We will first use an idea of Vinogradov in order to calculate the Fourier transformation of ψ and thus to get some exponential sums which are easier to treat. The following lemma will provide us with this Fourier transformation
Lemma 2.2 ([8, Theorem 1.8]). Let p(t) be a real function in [a, b]. Then for δ > 0 the estimation X a<n≤b ψ(p(n)) b − a δ + ∞ X ν=1 min δ ν2, 1 ν X a<n≤b e(νp(n)) holds.
The exponential sums arising in Lemma 2.2 are very similar to those in Nakai and Shiokawa [11]. Thus we will use their lemma for estimation.
Lemma 2.3 ([11, Lemma 6]). Let k, P and N be integers such that k ≥ 2, 2 ≤ N ≤ P . Let g(x) be real and have continuous derivatives up to the (k + 1)th order in [P + 1, P + N ]; let 0 < λ < 1/(2c0(k + 1)) and
λ ≤ g
(k+1)(x)
(k + 1)! ≤ c0λ (P + 1 ≤ x ≤ P + N ), or the same for −g(k+1)(x), and let
N−k−1+δ ≤ λ ≤ N−1 with 0 < δ ≤ k. Then P +N X n=P +1 e(g(n)) N1−η, where η = δ 16(k + 1)L, L = 1 + 1 4k(k + 1) + kR , R = 1 + $ log 1δk(k + 1)2 − log 1 − 1 k % .
If β0 is an integer we need some more delicate tools. In particular, we need a variation of the
above lemma for twice differentiable functions.
Lemma 2.4 ([15, Lemmas 4.4 and 4.7]). Let f (x) be a real function, twice differentiable, and let f00(x) ≥ λ > 0 throughout the interval (a, b), or the same for −f00(x). Then
X a<n≤b e(f (n)) = O 1 + |f 0(a) − f0(b)| √ λ + O (log(2 + |f0(a) − f0(b)|)) Now we have collected all the tools we need for the proof of Proposition 2.1.
Proof of Proposition 2.1. We start with an estimation of Ur,awith r such that (1.8) holds. Setting
δ = Nρ2 and ρ > 0, which we will choose later, an application of Lemma 2.2 yields
X N <n≤2N ψ p(n) qr+1 − a q N1−ρ2 + Nρ2 X ν=1 1 ν|S(N, r, ν)| , (2.2) where S(N, r, ν) = X N <n≤2N e νp(n) qr+1 . (2.3)
In order to properly estimate S(N, r, ν) we divide the proof up into two cases, according to whether β 6∈ Z or β ∈ Z.
• β is not an integer. We put k := bβc + 2 and g(x) := νq−(r+1)p(x). The idea is an
application of Lemma 2.3. Therefore we take a closer look at the (k + 1)st derivative of g. Since g(k+1)(x) ∼ ν qr+1α0β0(β0− 1) · · · (β0− k)x β0−k−1, we can bound it by λ < g (k+1)(x) (k + 1)! < c0λ (N < x ≤ 2N ) or similarly for −g(k+1)(x), where
λ = c ν qr+1N
β0−k−1.
In view of (1.8) we get that
Thus an application of Lemma 2.3 yields
S(N, r, ν) N1−η. (2.4)
Plugging this into (2.2) proves the proposition for this case.
• β is an integer. Since p is not a polynomial we get that there exists a βh6∈ Z. Let βh
be the biggest non-integer, i.e. β0, β1, . . . , βh−1∈ Z.
Now we divide the range of r up into three parts. In particular we distinguish between qr+1≤ Nβh−ρ, Nβh−ρ< qr+1≤ Nβ0−2+ρ, Nβ0−2+ρ< qr+1≤ Nβ0−1.
– qr+1 ≤ Nβh−ρ: As in the case of β
0 6∈ Z we want to apply Lemma 2.3. We set
g(x) := νq−(r+1)p(x) and since β0, . . . , βh−1∈ Z we get
λ < g
(β0+1)(x)
(k + 1)! < c0λ (N < x ≤ 2N ) or the same for −g(β0+1)(x), where
λ = c ν qr+1N
βh−β0−1.
Since qr+1≤ Nβh−ρwe get that
N−β0−1+ρ≤ λ ≤ Nβh−β0−1< N−1.
Thus an application of Lemma 2.3 yields S(N, r, ν) N1−η. (2.5)
– Nβh−ρ< qr+1 ≤ Nβ0−2+ρ: In this case we follow the proof of Theorem in [11] (cf.
Equation (12) of [11] and the following pages). In our case we set P = N , Q = N , and p := $ qr+1 ν β0−11 % N1−β0−11−ρ.
Imitating the proof in [11] we get S(N, r, ν) N1−2L1 p12β0(β0−1)(1−1/(β0−1))R q r+1 ν N + 1 2L1 + p. (2.6)
with L and R any positive integers. Since N ≤ Nβh−ρ≤ qr+1we get that νq−(r+1)N
ν. Thus the sum in (2.6) can be estimated by S(N, r, ν) N1−2L1 p 1 4Lβ0(β0−1)(1−1/(β0−1))Rν2L1 + N1− 1−ρ β0−1 N1− 1 2L+ ρ 2Lq r+1 4Lβ0(1−1/(β0−1))R+ N1−β0−11−ρ N1− 1 2L+ ρ 2L+β04Lβ0−2+ρ(1−1/(β0−1))R+ N1− 1−ρ β0−1 N1− 1 8L + N1− 1−ρ β0−1 (2.7)
where we have chosen R to be sufficiently large, such that β0(β0− 2 + ρ)(1 − 1/(β0−
1))R< 1.
– Nβ0−2+ρ< qr+1≤ Nβ0−1: Since β
0> βh> 1 and β0∈ Z we get that β0≥ 2. Thus
f00(n) > c ν qr+1N
β0−2=: λ > 0 (N < n ≤ 2N ).
We apply Lemma 2.4 in order to get S(N, r, ν) r ν qr+1N β0 2 N1− ρ 4 (2.8)
Plugging either (2.5), (2.7), or (2.5) into (2.2) proves the proposition for the case β06∈ Z.
3. Integral transform
The aim of this section is to transform the sum Ur,a into an integral Ir,a(N ) which is defined
by Ir,a(N ) := Z q−r−1p(2N ) q−r−1p(N ) ψ t − a q tβ01 −1dt. (3.1)
In particular we will show the following. Proposition 3.1. Let N be positive, then
Ur,a(N ) = 1 β0 α− 1 β0 0 q r+1 β0 Ir,a(N ) + O maxNβ1+1−β0, N23 (3.2)
The main tool will be the following Lemma.
Lemma 3.2 ([8, Theorem 1.5]). Let g in [a, b] (0 ≤ a < b) be a non-negative, strictly decreasing function with a continuous derivative in (a, b). If g−1(t) denotes the inverse function of g(t), then
X a<n≤b ψ(g(n)) − Z b a ψ(t)g0(t)dt − ψ(a)ψ(g(a)) = X g(a)<m≤g(b) ψ g−1(m) − Z g(b) g(a) ψ(t) g0(g−1(t))dt − ψ(b)ψ(g(b)).
The interesting integral is the last one. For the first integral we use partial integration and the sum is treated by the following lemma.
Lemma 3.3 ([8, Theorem 2.3]). Let p(t) be a real function in [a, b], twice continuously differen-tiable. Let p00(t) be monotonic and be either positive or negative throughout. Then
X a<n≤b ψ(p(n)) Z b a |p00(t)|13dt + 1 p|p00(a)|+ 1 p|p00(b)|.
Now we have collected all the tools for the proof of the lemma.
Proof of Proposition 3.1. In order to apply Lemma 3.2 we need that the function under consid-eration is monotonically decreasing. We guarantee this by rewriting the sum in (3.2) and then applying Lemma 3.2, which yields
X N <n≤2N ψ p(n) qr+1 − a q = X 0<n≤N ψ p (b2N c − n) qr+1 − a q = −q−r−1 Z N 0 ψ(t)p0(b2N c − t)dt + X q−r−1p(b2N c−N )−a q<m≤q−r−1p(b2N c)− a q ψ b2N c − p−1 qr+1 m + a q + qr+1 Z q−r−1p(b2N c) q−r−1p(b2N c−N ) ψt −aq p0(p−1(qr+1t))dt + O(1). (3.3)
We will write if both and hold. Since p is a pseudo-polynomial we can write p(t) = α0tβ0+ O(tβ1), which yields for the inverse of p
p−1(t) = t α0 β01 + Otβ1+1β0 −1 .
Implicitly calculating the derivatives gives p−10 (t) = 1 p0(p−1(t)) = 1 β0 α− 1 β0 0 t 1 β0−1+ O tβ1+1β0 −2 , p−100 (t) = − p 00(p−1(t)) (p0(p−1(t)))3 − β0− 1 β2 0 α− 1 β0 0 t 1 β0−2. (3.4) Now we define ψ1(x) :=R x
0 ψ(t)dt for x ∈ R and consider the three parts of (3.3). It is clear,
that ψ1(x) is continuous, bounded and piecewise continuously differentiable. Integration by parts
for the first integral in (3.3) yields the estimate (3.5) Z N 0 ψ(t)p0(b2N c − t)dt = ψ1(t)p0(b2N c − t)|N0 + Z N 0 ψ1(t)p00(b2N c − t)dt Nβ0−1.
In order to estimate the sum we define g(t) := b2N c − p−1(qr+1(t +a
q)) for t ≥ 0. For the range
q−r−1p(b2N c − N ) − aq < t ≤ q−r−1p(b2N c) − aq, we have t q−r−1Nβ0 and thus by (3.4)
|g00(t)| q(r+1)/β0t1/β0−2 q2(r+1)N1−2β0.
Then an application of Lemma 3.3 yields for the sum in (3.3) that (3.6) X q−r−1g(b2N c−N )−a q<m≤q−r−1g(b2N c)− a q ψ(g(m)) q−(r+1)/3N(β0+1)/3+ q−r−1Nβ0−12 N 2 3.
Now we aim for the main term which we extract from the second integral of (3.3). Noting that ψ(t) 1 we get with (3.4) that
qr+1 Z q−r−1p(b2N c) q−r−1p(b2N c−N ) ψt −aq p0(p−1(qr+1t))dt = qr+1 Z q−r−1p(b2N c) q−r−1p(b2N c−N ) ψ t −a q 1 β0 α− 1 β0 0 (q r+1t)β01−1 + O(qr+1t)β1+1β0 −2 dt = 1 β0 α− 1 β0 0 q r+1 β0 Z q−r−1p(b2N c) q−r−1p(b2N c−N ) ψ t −a q tβ01−1dt + O Z q−r−1p(b2N c) q−r−1p(b2N c−N ) ψ t −a q q(r+1) “β0+1 β0 −1 ” tβ1+1β0 −2dt ! = 1 β0 α− 1 β0 0 q r+1 β0 Ir,a(N ) + ( Z q−r−1p(N ) q−r−1p(b2N c−N ) + Z q−r−1p(b2N c) q−r−1p(2N ) ) ψ t − a q tβ01−1dt ! + O max(Nβ1+1−β0, log N ) = 1 β0 α− 1 β0 0 q r+1
β0 Ir,a(N ) + O (1) + O max(Nβ1+1−β0, log N )
(3.7)
Plugging (3.5), (3.6) and (3.7) into (3.3) proves the lemma. 4. Proof of Theorem 1.1
In the sections above we have collected all the tools we need along the road we will follow in order to proof Theorem 1.1. We split the sum in (1.5) up into dyadic parts
X n≤x f (bp(n)c) = X 1≤i≤log2x S(2−ix) + O(1), (4.1) where S(N ) := X N <n≤2N f (bp(n)c). (4.2)
The proof proceeds in three steps.
(1) In the first one, we use the q-additivity of f in order to count the occurrences of every digit separately. Therefore we will sum over all positions and exchange the sums in S(N ). As we will see, the resulting sums will be our well known Ur,a. Thus we will apply our
estimates from Section 2 and 3.
(2) Then we will give the integral resulting from the transformation in Section 3 the shape of a periodic function such as in the results of Delange [1] and Peter [12].
(3) Finally we will put all together in order to gain the desired result.
4.1. Rewriting the sum. The idea is to use the q-additivity of f and exchange the sum over n with the sum over the digits. Therefore we need a function which indicates if the digit at position r is a or not. The main idea is to extract the main term and write the rest as sum of ψ functions, i.e. Ir,a(x) = 1 q + ψ x qr+1− a + 1 q − ψ x qr+1− a q = ( 1 if dr(x) = a, 0 else. (4.3)
Now using the representations (4.3) and (1.7) in (4.2) we get that
S(N ) = X N <n≤2N X 0≤r≤R(2N ) q−1 X a=0 f (a)Ir,a(p(n)) = µfN logq(p(2N )) + q−1 X a=0 f (a) X 0≤r≤R(2N )
Ur,a+1(N ) − Ur,a(N ) + O(1).
(4.4)
The terms Ur,a occurring in (4.4) are treated in Section 2 and Section 3. Thus applying
Proposition 2.1 and 3.1 we gain
S(N ) = µfN X 0≤r≤R(2N ) 1 + q−1 X a=0 f (a) X 0≤r≤R(2N ))
Ur,a+1(N ) − Ur,a(N ) + O(1)
= µfN (bR(2N )c + 1) + O N1−η + 1 β0 α− 1 β0 0 q−1 X a=0 f (a) X 0≤r≤R(2N ) q(r+1)/β0(I r,a+1(N ) − Ir,a(N )) − 1 β0 α− 1 β0 0 q−1 X a=0 f (a) X qr+1≤Nβ0−1 q(r+1)/β0(I r,a+1(N ) − Ir,a(N )) + OmaxNβ1+1−β0, N23. (4.5)
4.2. Extraction of the periodic function. The goal of this intermediate section is the trans-formation of the integral Ir,a to give the result the shape as in the result of Delange [1] and
Peter [12].
Since we will often use the functions Jf,β0 and Ff,β0 defined in (1.4), we will write J := Jf,β0
and F := Ff,β0 for short. Then noting the definition of J together with partial integration as used
in (3.5) we get J (x) = q−1 X a=0 f (a) Z x 0 ψ t −a + 1 q − ψ t −a q tβ01−1dt 1 (x ≥ 0). (4.6)
Now we concentrate on the first sum over the r of the Ir,a in (4.4). By noting the definition of J in (1.4) we get that 1 β0 α− 1 β0 0 X r≤R(2N ) q(r+1)/β0 q−1 X a=0
f (a) (Ir,a+1− Ir,a)
= 1 β0 α− 1 β0 0 X r≤R(2N ) q(r+1)/β0J (q−r−1g(2N )) − X r≤R(2N ) q(r+1)/β0J (q−r−1g(N )) =: S1(N ) − S2(N ), N ≥ 1. (4.7)
Since p(t) = α0tβ0+ O(tβ1) and J (x) 1 by (4.6) we get on the one hand
S1(N ) = 1 β0 α− 1 β0 0 X r≤R(2N ) q(R(2N )−r+1)/kJ (qr−R(2N )−1g(2N )) = 1 β0 α− 1 β0 0 p(2N ) 1 β0q(1−{R(2N )})/β0 X r≤R(2N ) q−β0r J (q{R(2N )}+r−1) = 1 β0 2N q(1−{R(2N )})/β0 X r≤R(2N ) q−β0r J (q{R(2N )}+r−1) + O N X r≤R(2N ) q−β0r = 1 β0 2N q(1−{R(2N )})/β0 X r≥0 q−β0r J (q{R(2N )}+r−1) + O X r>R(2N ) q−β0r + O(1) = 2N F (R(2N )) − µf2N (1 − {R(2N )}) + O(1). (4.8)
For S2(N ) on the other hand we get
S2(N ) = S1 N 2 + 1 β0 α− 1 β0 0 X R(N )<r≤R(2N ) qr+1β0 J (q−r−1p(N )). (4.9)
For the second sum we have that R(N ) < r ≤ R(2N ), which implies q−r−1p(N ) < 1q. Thus connecting the representations in (4.3) and (1.4) we gain for 0 ≤ x < 1
q J (x) = Z x 0 q−1 X a=0 f (a) Ia(t) − 1 q t1k−1dt = −µfβ0x 1 β0. (4.10)
Now plugging (4.8) and (4.10) into (4.9) yields
S2(N ) = N F (R(N )) − µfN (1 − {R(N )}) − µfN (bR(2N )c − bR(N )c) + O(1).
(4.11)
Finally we consider the second sum over Ir,a in (4.5). Integration by parts yields for positive
integers a and r and N ≥ 1
Ir,a q “ 1−1 β0 ” (r+1) N1−β0. Thus we get 1 β0 α− 1 β0 0 q−1 X a=0 f (a) X qr+1≤Nβ0−1 q(r+1)/β0(I r,a+1(N ) − Ir,a(N )) 1. (4.12)
4.3. Putting all together. Plugging (4.8), (4.11) and (4.12) into (4.4) yields S(N ) = µf2N R(2N ) − µfN R(N ) + 2N F (R(2N )) − N F (R(N ))
+ OmaxNβ1+1−β0, N23, N1−η
. (4.13)
Plugging this into (4.1) and summing up over N = 2−ix for 1 ≤ i ≤ log 2x yields X n≤x f (bp(n)c) = X 1≤i≤log2x S(2−ix) + O(1) = µfx logq(p(x)) + Ff,β0 logq(p(x)) + O x 1−ε for an ε > 0, which proves the theorem.
Acknowledgment
I would like to thank J¨org Thuswaldner for drawing my attention to this problem, in general, and, in particular, to the paper of Manfred Peter [12].
References
[1] H. Delange, Sur la fonction sommatoire de la fonction“somme des chiffres”, Enseignement Math. (2) 21 (1975), no. 1, 31–47.
[2] P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R. F. Tichy, Mellin transforms and asymptotics: digital sums, Theoret. Comput. Sci. 123 (1994), no. 2, 291–314.
[3] A. O. Gel0fond, Sur les nombres qui ont des propri´et´es additives et multiplicatives donn´ees, Acta Arith. 13 (1967/1968), 259–265.
[4] B. Gittenberger and J. M. Thuswaldner, The moments of the sum-of-digits function in number fields, Canad. Math. Bull. 42 (1999), no. 1, 68–77.
[5] P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence, Constr. Approx. 21 (2005), no. 2, 149–179.
[6] P. J. Grabner, P. Kirschenhofer, H. Prodinger, and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, pp. 263– 271.
[7] P. Kirschenhofer, On the variance of the sum of digits function, Number-theoretic analysis (Vienna, 1988–89), Lecture Notes in Math., vol. 1452, Springer, Berlin, 1990, pp. 112–116.
[8] E. Kr¨atzel, Lattice points, Mathematics and its Applications (East European Series), vol. 33, Kluwer Academic Publishers Group, Dordrecht, 1988.
[9] C. Mauduit and J. Rivat, La somme de chiffres de carres, Preprint, submitted. [10] , R´epartition des fonctions q-multiplicatives dans la suite ([nc])
n∈N, c > 1, Acta Arith. 71 (1995), no. 2, 171–179.
[11] Y. Nakai and I. Shiokawa, A class of normal numbers, Japan. J. Math. (N.S.) 16 (1990), no. 1, 17–29. [12] M. Peter, The summatory function of the sum-of-digits function on polynomial sequences, Acta Arith. 104
(2002), no. 1, 85–96.
[13] I. Shiokawa, On the sum of digits of prime numbers, Proc. Japan Acad. 50 (1974), 551–554.
[14] J. M. Thuswaldner, The sum of digits function in number fields, Bull. London Math. Soc. 30 (1998), no. 1, 37–45.
[15] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.
(M. G. Madritsch) Department for Analysis and Computational Number Theory, Graz University of Technology, A-8010 Graz, Austria
