Mathematica - Fachgruppe Computeralgebra

Since 1996, numerous formulas of this same type
have been found for various constants [1, 8]. For example,
a similar formula was found that permits arbitrary-
position binary digits of π 2 to be calculated; here is a
formula for π 2 that permits arbitrary ternary (base-3)
digits of π 2 to be calculated:
π2 = 2 ∞ 1
27 k=0 729k
243 405 81 27
−
−
−
(12k + 1) 2 (12k + 2) 2 (12k + 4) 2 (12k + 5) 2
72
9
9
5
1
−
−
−
−
+
(12k + 6) 2 (12k + 7) 2 (12k + 8) 2 (12k + 10) 2 (12k + 11) 2
.
Sadly, it has recently been proven that there is no formula
of this type for π itself in other than a binary base [8,
Ch. 3].
Identification of Multiple ζ Constants
A large number of results has been found over the last
15 years using PSLQ in the course of research on multiple
zeta sums, such as those shown in Table 1. After
computing the numerical values of these constants, a
PSLQ program was used to determine if a given constant
satisfied an identity of a conjectured form. These
efforts produced numerous empirical evaluations and
suggested general results [2, 9]. Eventually, elegant
∞ k=1
∞ k=1
proofs were found for many of these specific and general
results. Three examples of PSLQ results that were
subsequently proven are given in Table 1. In the table,
ζ(t) =
∞
j −t
j=1
is the Riemann zeta function, and
Lin(x) =
∞
j=1
x j j −n
denotes the polylogarithm function.
1
1 2 −4 37
1 + 2 + · · · + k (k + 1) = 22680π6 − ζ2 (3)
1
1 3 −6 3 197 1
1 + 2 + · · · + k (k + 1) = ζ (3) + 24 ζ(9) + 2π2ζ(7) ∞
1
k=1 1 − 2
+ · · · + (−1)k+1 1
k
Ising integrals
One particularly fruitful application of these methods is
the evaluation of definite integrals, such as those that
arise in mathematical physics. For example, recently the
present authors, together with Richard Crandall, investigated
three classes of n-fold integrals, which arise in
Ising theory and also (in some cases) in quantum field
theory:
Cn := 4
∞
n! 0 · · · ∞
0
− 11
120π4ζ(5) − 37
7560π6ζ(3) 2 −3 (k + 1) = 4 Li5( 1 1
2 ) − 30 ln5 (2) − 17
32ζ(5) − 11
720 π4 ln(2) + 7
4 ζ(3) ln2 (2) + 1
18 π2 ln 3 (2) − 1
8 π2 ζ(3)
Table 1: Some multiple ζ identities found by PSLQ
1
( du1
n
2 u1
j=1 (uj+1/uj))
· · · dun
un ,
10
Dn := 4
∞
n! 0 · · · ∞
0
En = 2 1
0 · · · 1
0
2 ui−uj i