Bsplines and the Riemann's Zeta Function on
Integers
Alireza Entezari
entezari@cise.ufl.edu
Technical Report: REP2009474
CISE Department, University of Florida
Nov 08, 2008
1 Background and definitions
The Poisson's summation formula is:
1 2x
f(kT) Z (n ), (1)
kCZ nCZ
where f is the Fourier transform of f:
f(wu) I f(x) exp (iwx) dx.
The Poisson's summation formula holds for all integrable functions, given the
series on the right hand side is absolutely convergent.
The simplest (centered) Bspline, 3o, is the characteristic function of [1/2, 1/2).
Higher order Bsplines 3, are defined recursively by 3m = Pm1 po. Fourier
transform of a Bspline is:
(w) = sinc"m+(w), (2)
where sinc(t) := sin(t/2)/(t/2).
Lemma 1 (Partition of Unity).
m3,(k) 1
kcZ
Proof. Using Poisson's summation formula, setting T = 1, we have:
S3,m(k)= since + (2)7n) 1,
kCZ nCZ
since sin(rn) 0, for all n E Z and the only nonzero term in the right hand
side comes from n = 0: sinc(0) := 1 that agrees with the actual limit of sine at
0. O
1.1 Zeta function
The Riemann's Zeta function is defined as:
((x) (3)
nl
n 1
2 Relationship between /3m and (m n+ 1)
By playing around with the parameter T, one can recover parts of the series.
The linear Bspline can be related to the ((2) and generally, ,3m can be related
to ((m + 1). For the case of linear Bspline, by setting T = 2 in (1), we have:
3 Bi(2k) s sinc2(n7)
1C
kCZ neZ
2 + since nr)
nl
_1 + 0 sin2(n7)
2 (7/2)2 n2
1 4 sin 2(n)
_2 (/2 n2
nl
The last series is nearly ((2) except that sin2(n) zeroes out the even terms.
But it is, surely, convergent, since it is a subseries of ((2), so as the series with
remaining terms of ((2). The last series can be resolved, however, using ((2):
Ssin2(n) 1 17
n2 W2 2 (2n)2
n=l n=l n=l
n=1
((2) ((2)
4
4
4((2).
Therefore, we can relate 31 to ((2) by:
1 3
31(2k) 2 + ((2). (4)
kcZ
Since the support of 31 is [1, 1), the left hand side is 1 and we have:
((2) 2
6
((2) has been used in the so called Basel problem. Its inverse is the probability
that two randomly selected integers are relatively prime.
2.1 Result
Using the above approach, one can derive the exact values for all even integers
of C using evaluation of Bsplines on even integers. Wikipedia lists C(2), ((4)
and C(6), but, apparently, evaluation of C, in general is difficult and there are
papers for evaluation such as [1].
Is there a closedform solution for all Bsplines of odd order:
S 32m +(2k) =? (5)
kCZ
3 The difficult, but interesting case
The odd values of C, are interesting; for instance, C(3), known as Ap6ry's
constant [2] is a curious number that occurs in various physical problems. The
exact value of this constant is not known and it is an open problem whether
this number is transcendental.
To derive Ap6ry's constant using the Bspline approach, we shall focus on
the quadratic Bspline; employing the Poisson's sum (1), and T = 2, we have:
1 32(2k) 1 sincs(rn)
kcZ ntZ
1 sin3 (n")
2 (,(n)3
1 8 sin3(n )
2 + s3 ( ) even terms are zero.
2 + 8 n3
n= 1
The last series in the above derivation is more difficult to tackle since sin3 (n )
is alternating its sign for the nonzero terms.
sin n(n) \ sin ((2n +1)2)
S n3 (2n + 1)3
n=1 n=0
(1)
= w (6)
Y0(2n + 1)3 (6)
f 1 1
S(4n+ 1)3 (4n+ 3)
n=0 n=0
On the other hand:
((3) t 7
n=l
1 00
(2n + 1)3 (2n)3
(2n + 1)3 + (
n=0 n1
1 1
E + ((3).
n=(2n + 1)3 8
Hence, we have:
7 1
((3) 1 (2n + 1)3
n=0
00 00
1 1 +1O (7)
O(4 + )3 + (4n + 3)3
(7) and (6) are different and hence we can not resolve ((3) using this approach.
4 S.O.S
The question is: is there a more appropriate choice than T = 2? By choosing
different values like T 3/2, we get different subseries of ((3). Can we build
((3), perhaps, from multiple choices for T?
References
[1] Jonathan M. Borwein, David M. Bradley, and Richard E. Crandall. Compu
tational strategies for the Riemann zeta function. J. Comput. Appl. Math.,
121(12):247296, 2000. Numerical analysis in the 20th century, Vol. I, Ap
proximation theory.
[2] Wikipedia. Apery's constant wikipedia, the free encyclopedia,
http://en.wikipedia.org/wiki/Ap%C3%A9ry%27sconstant, 2008. [On
line; accessed 12November2008].
