Fractional Parts and their Relations to the Values of the Riemann Zeta Function

0.14 MB·

by  Ibrahim Alabdulmohsin

FRACTIONAL PARTS AND THEIR RELATIONS TO THE
VALUES OF THE RIEMANN ZETA FUNCTION
7
1
0
IBRAHIM M. ALABDULMOHSIN
2

n
a King Abdullah University of Science and Technology (KAUST),
J Computer, Electrical and Mathematical Sciences & Engineering Division

5 Thuwal, 23955-6900, Saudi Arabia
Tel: [phone]

]
T Email: [email protected]
N
.
h
t
a Abstract. Awell-knownresult,duetoDirichletandlatergeneralizedbyde
m laVall´ee-Poussin,expressesarelationshipbetweenthesumoffractionalparts
and the Euler-Mascheroni constant. In this paper, we prove an asymptotic
[
relationship between the summation of the products of fractional parts with

1 powersofintegersononehand,andthevaluesoftheRiemannzetafunction,
v onthe other hand. Dirichlet’sclassicalresultfallsasaparticular caseofthis
3 moregeneraltheorem.
8
8
4 1. Background
0
1. In 1849, Dirichlet established a relationship between the Euler-Mascheronicon-
0 stant γ =0.5772 and the average of fractional parts. More specifically, writing
7 [x] for the integr·a·l·(floor) part of the number x R and x = x [x] for its
1 ∈ { } −
fractional part, Dirichlet proved that [Pillichshammer, 2010, Lagarias,2013]:
:
v
n
i 1 n 1
X =1 γ+O (1.1)
r nXx=1nxo − (cid:16)√n(cid:17)
a
This surprising connection between γ and the average of fractional parts was, in
turn,usedbyDirichlettoprovethatthenumberofdivisorsofanintegernisofthe
order logn. The technique introduced by Dirichlet to prove these results is often
called the “hyperbola method”, which is a counting argument to the number of
lattice points that lie beneath a curve [Stopple, 2003, Lagarias,2013].
Theerrortermin(1.1)isknowntobepessimistic. Findingtheoptimalexponent
θ >0 such that:
n
n
(1 γ)n=O(nθ+ǫ),
xX=1nxo− −
for any ǫ>0 is knownas the Dirichlet divisor problem, which remains unsolvedto
this date. A well-known result of Hardy is that θ 1, which is conjectured to be
≥ 4
the true answer to this problem [Lagarias,2013].
1
2 FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION
In 1898, de la Vall´ee-Poussin generalized (1.1). He showed that for any integer
w N:
∈ n−1
w w n 1
=1 γ+O (1.2)
n xX=1nwx+1o − (cid:16)√n(cid:17)
As noted by de la Vall´ee-Poussin,this result is quite remarkablebecause the limit-
ing average of the fractional parts remains unchanged regardless of the arithmetic
progressionthat one wishes to use [Lagarias,2013].
More recently, Pillichshammer obtained a different generalization of Dirichlet’s
result. He showed that for any β >1:
√βn
n 1
xX=1nxβo=(1−γ1/β)√β n+O(cid:0)nβ+1(cid:1), (1.3)
whereγ isafamilyofconstantswhosefirsttermisγ =γ[Pillichshammer, 2010].
1/β 1
In this paper, we look into a different line of generalizing (1.1). Specifically, we
address the question of deriving the asymptotic expressions to summations of the
form:
n
n
f (n)= xs, (1.4)
s
x
xX=1n o
for non-negative integers s Z+. This is the summation of the products of frac-
∈
tional parts and powers of integers. Clearly, the case where s = 0 reduces to
Dirichlet’sclassicalresult. Interestingly,wewillshowthattheasymptoticbehavior
of this summation is connected to the values of the Riemann zeta function ζ(s),
andwewillrecoverDirichlet’sresultin(1.1)asaparticularcase. Morespecifically,
we prove that:
n
1 n 1 ζ(s+1) 1
xs = +O (1.5)
ns+1 xX=1nxo s − s+1 (cid:16)√n(cid:17)
The Riemann zeta function ζ(s) is a function of the complex variable defined in
(s)>1 by the absolutely converging series:
R
∞ 1
ζ(s)= , (1.6)
xs
Xx=1
andthroughoutthecomplexplaneCbyanalyticcontinuation. Itisameromorphic
function with a simple pole at s=1 with residue 1.
We conclude this section with two classical theorems that we will rely on in our
proofs.
Theorem 1 (AbelSummationFormula). Let a be a sequenceof complex numbers
x
and φ(x) be a function of class C1. Then:
x
a φ(x)=A(x)φ(x) A(t)φ(t)dt, (1.7)
x ′
−Z
1Xx n 1
≤ ≤
where A(x)= [x] a .
k=1 x
P
Theorem 2 (Euler-MaclaurinSummation Formula). We have:
n φ(x)=C+ nφ(t)dt+s−1 Bkφ(k)(n)+O(φ(s)(n)), (1.8)
Z k!
xX=1 1 Xk=1
FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION 3
for some constant C, where B = 1,B = 1,B =0,... are the Bernoulli numbers.
1 2 2 6 3
Theseresultscanbefoundinmanyplaces,suchas[Hardy, 1949,Lampret, 2001].
2. Notation
We will use the following notation:
[x] denotes the integral (floor) part of x and x = x [x] denotes the
• { } −
fractional part.
Ndenotesthesetofpositiveintegers,oftencalledthe natural numbers; Z+
• is the setofnon-negativeintegers;R isthe setofrealnumbers; C isthe set
of complex numbers.
(s) denotes the real part of s C.
• RS denotes the size (cardinality)∈of the set S.
• | |
3. The Fractional Transform
3.1. Overview. Thekeyinsightwewillemploytoderivetheasymptoticexpansion
of the function f (n) in (1.4) is that we can solve this problem indirectly by an-
s
sweringadifferent question,first. Specifically,wewillbeinterestedinthefollowing
function:
n 1
− n n
Φ (n)= xs (3.1)
s
xX=1hnxo−nx+1oi
More generally, when:
n 1
− n n
Φ(n)= φ(x), (3.2)
xX=1hnxo−nx+1oi
we will call Φ(n) is the fractional transform of φ(n). The reason Φ (n) allows us
s
to answer our original question is because:
n 1 n
− n n n
Φ (n)= xs = xs (x 1)s
s
xX=1hnxo−nx+1oi xX=1nxo(cid:16) − − (cid:17)
Byexpanding the right-handside using the binomialtheorem,we obtaina method
of solving our original question.
3.2. Prelimenary Results. Next, we present a few useful lemmas related to the
fractional transform defined above. Before we do this, we introduce the following
symbol:
n n
∂(n)= x N: , N= (3.3)
n ∈ hx+1 xi∩ 6 ∅o
In other words, ∂(n) is the set of positive integers that are less than n, and for
which the interval [n/(x+1),n/x] contains, at least, one integer. For instance,
2 ∂(5) because the interval [5/3,5/2] contains the integer two, whereas 3 / ∂(5)
∈ ∈
because the interval [5/3,5/4] lies strictly between the integer one and the integer
two.
Lemma 1.
n n n n n
= , N , (3.4)
nxo−nx+1o x(x+1) −(cid:12)hx+1 xi∩ (cid:12)
where S denotes the size (cardinality) of the set(cid:12)(cid:12)S. (cid:12)(cid:12)
| |
4 FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION
Proof. We have:
n n n n n n
=
nxo−nx+1o (cid:16)x − x+1(cid:17)−(cid:16)hxi−hx+1i(cid:17)
n n n n n n
= + = , N
x(x+1) −hxi hx+1i x(x+1) −(cid:12)hx+1 xi∩ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:3)
Lemma 2. If x √n and x ∂(n), then:
≥ ∈
n n n
= 1
{x}−{x+1} x(x+1) −
Proof. Because the interval [n/(x+1),n/x] can contain, at most, a unique integer
since:
n n n n
= <1
x − x+1 x(x+1) ≤ √n(√n+1)
This fact and Lemma 1 bot imply the statement of the lemma. (cid:3)
4. Main Results
We begin with the following lemma:
Lemma 3. For all s R:
∈
nǫ
n n
xs =O(nǫs), (4.1)
Xx=1hnxo−nx+1oi
whose the O() constant depends on s.
·
Proof. First, let us consider the following function:
w
n n
g (w)= x
n
xX=1hnxo−nx+1oi
Since:
w w w
n n n n
= x (x 1)=g (w)+ w
n
xX=1nxo xX=1nxo −xX=1nxo − nw+1o
we obtain:
w
n n
g (w)= w+ =O(w)
n
−nw+1o xX=1nxo
More precisely, we conclude that:
x g (x) x
n
− ≤ ≤
FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION 5
Using this fact and the Abel summation formula in Theorem 1:
nǫ nǫ
n n n n
xs = xxs−1
xX=1hnxo−nx+1oi xX=1hnxo−nx+1oi
nǫ
=nǫ(s−1)gn(nǫ) (s 1) gn(t)ts−2dt
− − Z
1
nǫ
nǫs (s 1) g (t)ts 2dt
n −
≤ − − Z
1
nǫ s 1
nǫs+ s 1 ts−1dt 1+ | − | nǫs
≤ | − |Z1 ≤(cid:16) s (cid:17)
Similarly:
√n
n n s 1
xs 1+ | − | nǫs
xX=1hnxo−nx+1oi ≥−(cid:16) s (cid:17)
Therefore, the statement of the lemma follows. (cid:3)
Now, we are ready to prove our first main result.
Theorem 3. For any s>1:
n 1
1 − n n 1 1
xs = 1+ζ(s)+O
ns Xx=1hnxo−nx+1oi s−1 − (cid:16)√n(cid:17)
Proof. We split the sum into two parts:
√n n 1
n n − n n
Φ (n)= xs+ xs (4.2)
s
xX=1hnxo−nx+1oi x=X√n+1hnxo−nx+1oi
ThefirsttermisO(n2s)asprovedinthepreviouslemma,whichisO(ns−12)when
s 1. Next, we examine the second term. We have by Lemma 2:
≥
n−1 n n n−1 nxs n−1 n n
xs = , N xs
X hnxo−nx+1oi X x(x+1) − X (cid:12)hx+1 xi∩ (cid:12)
x=√n+1 x=√n+1 x=√n+1(cid:12) (cid:12)
(cid:12) (cid:12)
n−1 nxs √n−1 n s
=
x=X√n+1x(x+1) − xX=2 hxi
n−1 nxs √n−1 n s √n−1 n s
= +
x=X√n+1x(x+1) − xX=2 (cid:16)x(cid:17) xX=2 nxo
n−1 nxs √n−1 1
= ns +O(√n)
x(x+1) − xs
x=X√n+1 xX=2
Using the fact that for (s)>0 [Cvijovic and Srivastava, 2012]:
R
w 1 w1−s ∞ t
ζ(s)= + s { } dt,
xs s 1 − Z ts+1
xX=1 − w
6 FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION
we conclude that:
w 1 w1 s
= − +ζ(s)+O(w s)
−
xs 1 s
xX=1 −
Alternatively, the error term O(w s) in the above expression can be derived from
−
the Euler-Maclaurinsummation formula in Theorem 2. Hence:
√n 1 1−s 1+s
ns − 1 =ns n 2 +ζ(s) 1+O(n−2s) =(ζ(s) 1)ns+ n 2 +O(ns2),
xX=2 xs (cid:16)1−s − (cid:17) − 1−s
Finally, we look into the remaining term:
n−1 xs−1 n−1 xs−2
n =n
x=X√n+1x+1 hx=X√n+11+ x1i
n 1
− 1 1
=n xs−2 1 +
hx=X√n+1 (cid:0) − x x2 −···(cid:1)i
n 1
−
=n xs 2 xs 3+
− −
h X − ···i
x=√n+1
=n ns−1 n12(s−1) n21(s−2)+ns−2 +O(n12(s−1)+ns−2)
hs 1 − s 1 − 2 i
− −
= ns + n12(s+1) +O(ns−21),
s 1 1 s
− −
fors 1. This followsfromthe Euler-Maclaurinsummationformula. Morespecif-
≥
ically, we have for u 2:
≥
n−1xs−u =C(u)+ ns−u+1 ns−u + m s−u Bk ns−u−k+1+O(ns−u−m)
s u+1 − 2 (cid:18)k 1(cid:19) k
xX=1 − Xk=2 −
√n s−u+1 s−u m
xs−u =C(u)+ n 2 + n 2 + s−u Bk ns−u−2k+1 +O(ns−u2−m),
s u+1 2 (cid:18)k 1(cid:19) k
xX=1 − Xk=2 −
for some constant C(u) that is independent of n.
Putting everything together, we conclude that for s>1:
n 1
− n n xs = 1 1+ζ(s) ns+O ns−21 ,
xX=1hnxo−nx+1oi hs−1 − i (cid:0) (cid:1)
which is the statement of the theorem. (cid:3)
Theorem 3 is illustrated in Fig 1. Clearly, this theorem generalizes Dirichlet’s
result, as promised earlier, because:
1
lim ζ(s) = γ
s 1ns 1 − o −
→ −
and the fact that:
n 1 n 1 n
− n n − n n
x= x (x 1) =
xX=1hnxo−nx+1oi xX=1nxo(cid:0) − − (cid:1) xX=1nxo
FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION 7
450 s=1 4x 105 s=2 3x 108 s=3
400 3.5
2.5
350 3
300 2
2.5
250
2 1.5
200
1.5
150 1
100 1
0.5
50 0.5
0 0 0
0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000
n
Figure 1. A comparison between the values of the fractional
transform of xs (marked in blue) and the asymptotic expression
derived in Theorem 3 (marked in red). The x-axis is n while the
y-axis is Φ(n). The left, middle, and right figures correspond to
s=1, s=2 and s=3 respectively.
Now, we are ready to derive the asymptotic expression of the function f (n)
s
given in (1.4).
Theorem 4. For all s Z+:
∈
n
1 n 1 ζ(s+1) 1
xs = +O
ns+1 xX=1nxo s − s+1 (cid:16)√n(cid:17)
Proof. Let f (n) be as defined in (1.4). Then, writing by Theorem 3:
s
n 1
1 +1 ζ(s) ns+O(ns−21)= − n n xs
hs−1 − i xX=1hnxo−nx+1oi
n 1 n 1 s
− n − n s
= xs (x 1)s = ( 1)k+1 xs k
−
xX=1nxo(cid:16) − − (cid:17) xX=1nxoXk=1 − (cid:18)k(cid:19)
s s
s s
= ( 1)k+1 f (n)=sf (n)+ ( 1)k+1 f (n)
s k s 1 s k
Xk=1 − (cid:18)k(cid:19) − − Xk=2 − (cid:18)k(cid:19) −
Hence for s N:
∈
1 ζ(s)
fs 1(n)= ns+O(ns−21),
− hs 1 − s i
−
which implies the statement of the theorem. (cid:3)
5. Conclusion
Inthispaper,wegeneralizedDirichlet’sclassicalresultontheconnectionbetween
the Euler-Mascheroni constant and the average of fractional parts. Our theorem
reveals that the fractional parts are, in general, connected to the values of the
Riemannzetafunctionζ(s). Hence,allofthevaluesofζ(s)atpositiveintegerscan
be expressedas limiting averagesofproducts offractionalpartswith the powersof
positive integers.
8 FRACTIONAL PARTS AND THEIR RELATIONS TO THE ZETA FUNCTION
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