Download Full text

ON THE FIBONACCI NUMBERS AND THE DEDEKIND SUMS
Zhang Wenpeng and Yi Yuan
The Research Center for Science, Xi'an Jiaotong University,
Xi'an, Shaanxi, Peoples Republic of China
(Submitted July 1998-Final Revision April 1999)
1. INTRODUCTION
As usual, the Fibonacci sequence F = (F„) is defined by F0 = 0, Fx = 1, and by the secondorder linear recurrence sequence Fn+2 =Fn+l+Fn for n>0. This sequence has many important
properties, and it has been investigated by many authors. In this paper we shall attempt to study
the distribution problem of Dedekind sums for Fibonacci numbers and obtain some interesting
results. For convenience, we first introduce the definition of the Dedekind sum S(h, q). For a
positive integer q and an arbitrary integer h, we define
s
where
™4M)
[x -[*]-•£
lo
«*»=
ah
9
if x is not an integer;
ifxis an integer.
The various arithmetical properties of S(h9 k) can be found in [3], [4], and [6]. About Dedekind
sums and uniform distribution, Myerson [5] and Zheng [7] have obtained some meaningful
conclusions. However, it seems that no one has yet studied the mean value distribution of
$(FmFn+l), at least we have not found expressions such as HS(Fn,Fn+l) in the literature. The
main purpose of this paper is to study the mean value distribution of S(Fn,Fn+l) and present a
sharper asymptotic formula. That is, we shall prove the following main theorem.
Theorem; Let m be a positive integer, then we have
1
2m
48
a
where a = ^~-, C(m) is a constant depending only on the parity of m, i.e.,
n=l
\«+l
i
1
C(«) =
+_Ly(iT
1 °°
1
To" 2«i
1
if m is an even number;
12 n=l
^2n^2n+l
^2n+l^2n+2
,_Ly<^
12 a
w+l
if m is an odd number.
2* SOME LEMMAS
To complete the proof of the theorem, we need the following two lemmas.
Lemma 1: Let m be a positive integer, then we have
S F
( m> Fm+l) +
2000]
1
1
m-l
UFnm+l
12
r
r
m m+l
1
F F
m-2
• + (-!>
1
^ 3 .
223
ON THE FIBONACCI NUMBERS AND THE DEDEKIND SUMS
Proof: It is clear that (Fm, Fm+l) = 1, iw = 1,2,3,..., so, from the reciprocity formula of
Dedekind sums (see [2] or [3]), we get
= F™+F^ + l-±
$(Fm,Fm+d+S(Fm+hFm)
l2F
(1)
mFm+l
By the recursion relationship Fm+l = Fm + Fm_l for /w>0, we have S(Fm+l,Fm) = S(Fm_uFm).
Thus,
s F
( m>
F
m+i) + S(Fm_h
F2+F2 , + 1 1
Fm)
l2F
4
F
m m+l
1 (F*-i
=
l 2
\
F
\ ( F
12 { rm+l
, ^
,•
F
F
F
T
^
m
m+l
Fm+l |
1
tm
1
m m+l)
1
L.
^m^m+i
4
F
6
1
m-\
=
12
^*A+1
12
12
^+1
m-2
^n '
so that
S(F^,Fm)
1
l
2iE
+
1
S F
m-lFm
w^+l
1
1
2F
l^mKn-l
F
( m-2>
l2F
m-l) + 1
1
F
l2F
^ m-l m
0
^
3
(-1) m-2 S(FUF2)
F
+
m-2 m-l
12F2
It is clear that S(Fl9 F2) = S(l, 1) = 0 and F0 = 0, so we obtain
S(Fm,Fm+1) +
1
Fm-i
1
m-2
• + • •• + (-!)'
1
12F2F3
This concludes the proof of Lemma 1.
Lemma 2: Let m be a positive integer, then we have
where a = ^^-.
Proof: From the second recursion relationship for F„, we can easily deduce that
1
''i+V5Y fi-VT^
®
and aFm = Fm+l +
From these identities, we get
f F„
\^aFn
"= lF"* W +*l a^F
*"* = l
W
W
J
l^F^+gy
,
M+l
= a±m+Y
*->
a^
^ «=1
f•* «+l
\W+I
ar.s-i
n=\ Ai+1
«+i
tn +
(i)1
«=i A?+i
-te>
This completes the proof of Lemma 2.
224
[JUNE-JULY
ON THE FIBONACCI NUMBERS AND THE DEDEKIND SUMS
3. PROOF OF THE THEOREM
In this section we shall complete the proof of-the theorem. First, let m b e a positive integer,
then from (1) we have
F
1
S F
( m> Fm+l) + S(Fm+h Fm) =
m+l I •* m
.
*_
12 I F
'F
'FF
14
or
^.,F„1)+^„^4(^+A_+^-)-I
and
Id[S(Fn,F„+l) + S(F„_l,F„)] = ±Y, 17
12:«=1 L w+l
n=l
F«
FF
6'
-'w'/H-l.
Noting that
S(F0,Fl) = S(0,l) = 0 and F0 = 0
so that
m
m
1
1 -m F
F
' w =i
' w=2 * »
x
x
w+l
1
m
^
w =i
1
fti
-* n* n+l
hence,
2ZS(F„,Fn+l) = S(Fm,Fm+l) + 1 ^,-1 , 1i y _ L + l f
F„
2m + l
12
12 F_,
6±{F
w+1
«=1 w+1
' = l »-* «+l
rtl ' 12f-/^F.
v
«=1
J
W
z
(2)
Applying (2), Lemma 1, and Lemma 2, we obtain
2£S(Fn,Fn+l)
M=I
1
= ±
12
__
1
j
|_ / _ i \ m - 2
1
\w+l
4zl w+ |;(iL +0 r i
+-
2/w
a'
«=1 Ai+1
2m + l
12 '
12„=1F„FW+1
Ifm is an even number, then from the above we have
Z
„=\
1 Z
^
n=\
r
2nr2n+l
- (V5-D 2 m l if
-
48
W+
ljL
i
n=\
r
n+\
\
a
Jim
\n+l
, if(r
12£lF 2 „F 2 n + 1 + 12^F„ + I
,fn
l C
+C
'U 2 '"J'
If HI is an odd number, then
w
=l
^^
48
1 Z
w =l
r
2n+lr2n+2
1Z,
W+
12^F 2n+1 F 2 „ +2 + 12 £ F„+1
n=l
^w+1
V
+ty
a
U2mJ'
This completes the proof of the theorem.
2000]
225
ON THE FIBONACCI NUMBERS AND THE DEDEKIND SUMS
ACKNOWLEDGMENTS
The authors express their gratitude to the anonymous referee for very helpful and detailed
comments.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Tom M. Apostol. Introduction to Analytic Number Theory. New York: Springer-Verlag,
1976.
Tom M. Apostol. Modular Functions andDirichlet Series in Number Theory. New York:
Springer-Verlag, 1976.
L. Carlitz. "The Reciprocity Theorem for Dedekind Sums." Pacific J. Math. 3 (1953):52327.
L. J. Mordell. "The Reciprocity Formula for Dedekind Sums." Amer. J. Math. 73 (1951):
593-98.
G. Myerson. "Dedekind Sums and Uniform Distribution." J. Number Theory 28 (1991):
1803-07.
H. Rademacher. "On the Transformation of log^(r)." J. Indian Math. Soc. 19 (1955):2530.
Z. Zheng. "Dedekind Sums and Uniform Distribution (mod 1)." Acta Mathematica Sinica
11 (1995):62-67.
AMS Classification Numbers: 11B37, 11B39
226
[JUNE-JULY