B-splines and the Riemann's Zeta Function on
Technical Report: REP-2009-474
CISE Department, University of Florida
Nov 08, 2008
1 Background and definitions
The Poisson's summation formula is:
f(kT) Z- (n ), (1)
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) B-spline, 3o, is the characteristic function of [-1/2, 1/2).
Higher order B-splines 3, are defined recursively by 3m = Pm-1 po. Fourier
transform of a B-spline is:
(w) = sinc"m+(w), (2)
where sinc(t) := sin(t/2)/(t/2).
Lemma 1 (Partition of Unity).
Proof. Using Poisson's summation formula, setting T = 1, we have:
S3,m(k)= since + (2)7n) 1,
since sin(rn) 0, for all n E Z and the only non-zero term in the right hand
side comes from n = 0: sinc(0) := 1 that agrees with the actual limit of sine at
1.1 Zeta function
The Riemann's Zeta function is defined as:
2 Relationship between /3m and (m n+ 1)
By playing around with the parameter T, one can recover parts of the series.
The linear B-spline can be related to the ((2) and generally, ,3m can be related
to ((m + 1). For the case of linear B-spline, by setting T = 2 in (1), we have:
3 Bi(2k) s- sinc2(n7)
2 + since nr)
_1 + 0 sin2(n7)
2 (7/2)2 n2
1 4 sin 2(n)
_2 (/2 n2
The last series is nearly ((2) except that sin2(n-) zeroes out the even terms.
But it is, surely, convergent, since it is a sub-series 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
Therefore, we can relate 31 to ((2) by:
31(2k) 2 + ((2). (4)
Since the support of 31 is [-1, 1), the left hand side is 1 and we have:
((2) has been used in the so called Basel problem. Its inverse is the probability
that two randomly selected integers are relatively prime.
Using the above approach, one can derive the exact values for all even integers
of C using evaluation of B-splines 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 .
Is there a closed-form solution for all B-splines of odd order:
S 32m +(2k) =? (5)
3 The difficult, but interesting case
The odd values of C, are interesting; for instance, C(3), known as Ap6ry's
constant  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 B-spline approach, we shall focus on
the quadratic B-spline; employing the Poisson's sum (1), and T = 2, we have:
1 32(2k) -1 sincs(rn)
1 sin3 (n")
1 8 sin3(n )
2- + s3 ( ) even terms are zero.
2 + 8 n3
The last series in the above derivation is more difficult to tackle since sin3 (n )
is alternating its sign for the non-zero terms.
sin n(n) \ sin ((2n +1)2)
S n3 (2n + 1)3
= w (6)
Y-0(2n + 1)3 (6)
f 1 1
S(4n+ 1)3 (4n+ 3)
On the other hand:
((3) t 7
(2n + 1)3 (2n)3
(2n + 1)3 + (
E + -((3).
n=-(2n + 1)3 8
Hence, we have:
((3) 1 (2n + 1)3
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.
The question is: is there a more appropriate choice than T = 2? By choosing
different values like T 3/2, we get different sub-series of ((3). Can we build
((3), perhaps, from multiple choices for T?
 Jonathan M. Borwein, David M. Bradley, and Richard E. Crandall. Compu-
tational strategies for the Riemann zeta function. J. Comput. Appl. Math.,
121(1-2):247-296, 2000. Numerical analysis in the 20th century, Vol. I, Ap-
 Wikipedia. Apery's constant -wikipedia, the free encyclopedia,
http://en.wikipedia.org/wiki/Ap%C3%A9ry%27sconstant, 2008. [On-
line; accessed 12-November-2008].