Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
An Odd Variant of
Multiple Zeta Values
Michael E. Hoffman
U. S. Naval Academy
Number Theory Talk
Max-Planck-Institut für Mathematik, Bonn
20 June 2014
ME Hoffman
Multiple t-Values
Outline
Multiple
t-Values
ME Hoffman
Outline
1 Introduction
Introduction
MtVs as
images of
quasisymmetric
functions
2 MtVs as images of quasi-symmetric functions
3 Iterated integrals
Iterated
integrals
Calculations
and
conjectures
4 Calculations and conjectures
Depth 2
results
5 Depth 2 results
ME Hoffman
Multiple t-Values
Introduction
Multiple
t-Values
ME Hoffman
For positive integers a1 , . . . , ak with a1 > 1 we define the
corresponding multiple t-value (MtV) by
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
t(a1 , a2 , . . . , ak ) =
X
1
.
· · · nkak
n a1 n a2
n1 >n2 >···>nk ≥1, ni odd 1 2
Except for the specification that the ni are odd, this is just the
usual definition of the multiple zeta value ζ(a1 , . . . , ak ). We
say t(a1 , . . . , ak ) has depth k and weight a1 + · · · + ak . MtVs
of depth 1 are simply related to values of the Riemann zeta
function:
X
X 1
1
=
ζ(a)−
= 1 − 2−a ζ(a). (1)
t(a) =
a
a
n
n
n > 0 even
n > 0 odd
ME Hoffman
Multiple t-Values
Introduction cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
While multiple zeta values (for depth ≤ 2) have a history going
back to Euler, t-values can only claim a history of 98 years. In
1906 N. Nielsen showed that
n−1
X
t(2i)t(2n − 2i) =
i=1
2n − 1
,
2
which parallels the result Euler gave for zeta values:
n−1
X
ζ(2i)ζ(2n − 2i) =
i=1
ME Hoffman
Multiple t-Values
2n + 1
.
2
Introduction cont’d
Multiple
t-Values
ME Hoffman
Outline
It turns out that the theory of multiple t-values is in some ways
directly parallel to that for multiple zeta values, and in other
ways completely different. The MtVs share with the MZVs the
“harmonic algebra” (AKA “stuffle”) multiplication, e.g., just as
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
ζ(2)ζ(3, 1) = ζ(2, 3, 1) + ζ(3, 2, 1) + ζ(3, 1, 2) + ζ(5, 1) + ζ(3, 3)
we have
t(2)t(3, 1) = t(2, 3, 1) + t(3, 2, 1) + t(3, 1, 2) + t(5, 1) + t(3, 3).
From this (and equation (1)) follows, e.g.,
π 2n
,
t(2, . . . , 2) =
| {z }
2(2n)!
n
ME Hoffman
π 2n
cf. ζ(2, . . . , 2) =
.
| {z }
(2n + 1)!
n
Multiple t-Values
Introduction cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Where MZVs and MtVs differ radically is in their representation
as iterated integrals. The iterated integral representation of
MZVs leads to a second algebra structure (shuffle product) and
the duality theorem, but the iterated integral representation of
MtVs is not nearly so nice. This is already apparent in weight
3: while for MZVs one has the first instance of duality
ζ(2, 1) = ζ(3)
(discovered by Euler and many others since), for MtVs one has
1
t(2, 1) = − t(3) + t(2) log 2.
2
ME Hoffman
Multiple t-Values
Introduction cont’d
Multiple
t-Values
MtVs as
images of
quasisymmetric
functions
Nevertheless, the iterated itegral represetation for MtVs does
lead to a formula expressing any MtV as a sum of alternating
MZVs. Thanks to the Multiple Zeta Value Data Mine project
of Blümlein, Broadhurst and Vermaseren, extensive tables
expressing alternating MZVs in terms of a (provisional) basis
exist. Our calculations using these tables have led us to the
following conjecture.
Iterated
integrals
Conjecture
Calculations
and
conjectures
The dimension of the rational vector space spanned by MtVs of
weight n ≥ 2 is Fn , the nth Fibonacci number.
ME Hoffman
Outline
Introduction
Depth 2
results
The corresponding conjecture for MZVs reads the same, but
with Fn replaced by Pn , the nth Padovan number (i.e., P0 = 1,
P1 = 0, P2 = 1, Pn = Pn−2 + Pn−3 ).
ME Hoffman
Multiple t-Values
MtVs as Multiple Hurwitz Zeta Functions
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
In the literature there are several results about multiple
t-values, but these are usually discussed in the context of
multiple Hurwitz zeta values, defined by
ζ(a1 , . . . , ak ; p1 , . . . , pk ) =
X
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
n1 >···>nk ≥1
(n1 + p1
) a1
1
.
· · · (nk + pk )ak
It is evident that
1
1
t(a1 , . . . , ak ) = 2−a1 −···−ak ζ(a1 , . . . , ak ; − , . . . , − ).
2
2
ME Hoffman
Multiple t-Values
Quasi-Symmetric Functions
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Consider the algebra Q[[x1 , x2 , . . . ]] of formal power series in a
countable set of commuting generators x1 , x2 , . . . . This has a
graded subalgebra P consisting of those series of bounded
degree (where each xi has degree 1). An element of f ∈ P is a
quasi-symmetric function if
coefficient of xia11 xia22 · · · xiakk in f =
coefficient of x1a1 x2a2 · · · xkak in f
Calculations
and
conjectures
Depth 2
results
whenever i1 < i2 < · · · < ik . The set of such f is a subalgebra
QSym of P, called the quasi-symmetric functions.
ME Hoffman
Multiple t-Values
Quasi-Symmetric Functions cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
As a vector space, QSym has the basis consisting of monomial
quasi-symmetric functions M(a1 ,...,ak ) , where (a1 , . . . , ak ) is a
composition (ordered sequence) of positive integers and
X a
M(a1 ,...,ak ) =
xi11 · · · xiakk .
i1 <···<ik
QSym has as a subalgebra the symmetric functions Sym; the
symmetric functions have the vector space basis
X
xia11 · · · xiakk
ma1 ,...,ak =
i1 , . . . , ik distinct
indexed by partitions (unordered sequences) of positive
integers. Note that, e.g.,
m2,2,1 = M(2,2,1) + M(2,1,2) + M(1,2,2) .
ME Hoffman
Multiple t-Values
Quasi-Symmetric Functions cont’d
Multiple
t-Values
ME Hoffman
Outline
In fact, QSym is a polynomial algebra on certain of the MI . We
order compositions lexicographically, i.e,
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
(1) < (1, 1) < (1, 2) < (1, 2, 1) < · · ·
and call a composition I Lyndon if I < K for any proper
factorization I = JK : e.g., (1, 3) is Lyndon but (1, 1) and
(1, 2, 1) aren’t. Then QSym is the polynomial algebra on the
MI with I Lyndon (Malvenuto and Reutenauer, 1995). Note
that the only Lyndon composition ending in 1 is (1). Let
QSym0 be the subalgebra of QSym generated by MI for
I 6= (1) Lyndon, so that QSym = QSym0 [M(1) ].
ME Hoffman
Multiple t-Values
Quasi-Symmetric Functions and MtVs
Multiple
t-Values
The following result is well-known (Hoffman 1997).
ME Hoffman
Theorem
Outline
There is a homomorphism ζ : QSym0 → R sending 1 to 1 and
M(a1 ,a2 ,...,ak ) to ζ(ak , . . . , a2 , a1 ) for ak ≥ 2.
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
This is paralleled by
Theorem
There is a homomorphism θ : QSym0 → R sending 1 to 1 and
M(a1 ,a2 ,...,ak ) to t(ak , . . . , a2 , a1 ) for ak ≥ 2.
It follows that if any linear combination of t-values is the image
of a symmetric function, then it is actually expressible as a
rational polynomial in the depth 1 t-values, e.g.,
t(2, 3) + t(3, 2) = t(2)t(3) − t(5).
ME Hoffman
Multiple t-Values
t-Values of Repeated Arguments
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
In particular, any t-value of repeated arguments is a rational
polynomial in the t(i). If we write {k}n for k repeated n times,
then we have the following.
Theorem
For integer k ≥ 2, let Zk (x) be the generating function
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Zk (x) = 1 +
∞
X
ζ({k}i )x ik .
i=1
Then
1+
∞
X
t({k}i )x ik =
i=1
ME Hoffman
Multiple t-Values
Zk (x)
.
Zk ( x2 )
t-Values of Repeated Arguments cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Now the generating functions Zk (x) were studied in some detail
back in the 1990’s: see Broadhurst, Borwein and Bradley 1997.
From their results and the preceding theorem we have, e.g.,
t({4}n ) =
π 4n
π 6n
,
t({6}
)
=
,
n
22n (4n)!
(8n)(6n − 1)!
π 10n (L10n + 1)
t({10}n ) =
,
(32n)(10n − 1)!
where in the last formula Ln is the nth Lucas number, i.e.,
L0 = 2, L1 = 1 and Ln = Ln−1 + Ln−2 .
ME Hoffman
Multiple t-Values
Iterated Integral Representations
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Call a sequence of (a1 , . . . , ak ) of positive integers admissible if
a1 > 1. Now if
dt
dt
dt
, ω−1 =
,
ω0 = , ω1 =
t
1−t
1+t
then there is the well-known representation
Z 1
ζ(a1 , . . . , ak ) =
ω0a1 −1 ω1 · · · ω0ak −1 ω1
0
for any admissible sequence (a1 , . . . , ak ), which extends as
follows. If (a1 , . . . , ak ) is admissible and 1 , . . . , k ∈ {−1, 1},
then
Z 1
ω0a1 −1 ω1 · · · ω0ak −1 ωk =
0
1 · · · k ζ(1 a1 , 1 2 a2 , 2 3 a3 , . . . , k−1 k ak )
ME Hoffman
Multiple t-Values
Iterated Integrals cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
where for ai ∈ {−1, 1} we define the alternating MZV by
X
sgn(a1 )n1 · · · sgn(ak )nk
ζ(a1 , . . . , ak ) =
.
|a |
|a |
n1 1 · · · nk k
n1 >···>nk ≥1
For multiple t-values there is also an iterated integral
representation, but it is not so simple.
Proposition
For any admissible sequence (a1 , . . . , ak ),
Calculations
and
conjectures
Z
1
t(a1 , . . . , ak ) =
0
Depth 2
results
a
ω0a1 −1 Ωa · · · ω0k−1
−1
Ωa ω0ak −1 Ωb
where
Ωa =
tdt
1
= (ω1 − ω−1 ),
2
1−t
2
ME Hoffman
Ωb =
Multiple t-Values
dt
1
= (ω1 + ω−1 ).
2
1−t
2
Iterated Integrals cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Expanding out the preceding result gives
Theorem
For an admissible sequence (a1 , . . . , ak ), the multiple t-value
t(a1 , . . . , ak ) is given by
1 X
1 · · · k ζ(1 a1 , . . . , k ak ),
2k ,...,
1
k
where the sum is over the 2k k-tuples (1 , . . . , k ) with
i ∈ {−1, 1}.
ME Hoffman
Multiple t-Values
The Multiple Zeta Value Data Mine
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Using the preceding result, we can expand any MtV into a sum
of alternating MZVs. For example,
1
ζ(2, 3, 1) − ζ(2̄, 3, 1) − ζ(2, 3̄, 1) − ζ(2, 3, 1̄)
8
+ ζ(2̄, 3̄, 1) + ζ(2̄, 3, 1̄) + ζ(2, 3̄, 1̄) − ζ(2̄, 3̄, 1̄) ,
t(2, 3, 1) =
where we have used the usual convention of writing n̄ instead
of −n in the exponent string. Using results of the Multiple
Zeta Values Data Mine project of Blümlein, Broadhurst and
Vermaseren, we expand each of the alternating MZVs in this
sum to obtain
t(2, 3, 1) = −
3
1
1
2
t(6) −
t(3)2 − t(2)ζ(3̄, 1) + ζ(5̄, 1)
21
196
2
4
1
4
− t(5) log 2 + t(2)t(3) log 2.
2
7
ME Hoffman
Multiple t-Values
Tables and Conjectures
Multiple
t-Values
ME Hoffman
Similarly we have obtained formulas for all multiple t-values of
weight ≤ 7. In our tables we have
4
1
t(2, 3) = − t(5) + t(2)t(3).
2
7
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
This and many similar instances have led us to the following
conjecture.
Conjecture
Every multiple t-value admits a representation as a polynomial
in the t(i), i ≥ 2, log 2, and selected alternating MZVs in such
a way that
∂t(s, 1)
= t(s).
∂ log 2
for any admissible sequence s.
ME Hoffman
Multiple t-Values
The Fibonacci Conjecture
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
The second conjecture is based on computations of the rank of
the set of multiple t-values of a given weight. Based on our
tables through weight 7, we have computed the ranks (starting
in weight 2) as
1, 2, 3, 5, 8, 13
which leads to the obvious conjecture
Conjecture
The dimension of the rational vector space spanned by
weight-n multiple t-values is Fn .
ME Hoffman
Multiple t-Values
Fibonacci Conjecture cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
Of course the existing evidence for the Fibonacci Conjecture is
rather thin. The Multiple Zeta Value Data Mine actually has
publically available files on alterating MZVs for weights
through 14, so there is plenty of testing yet to do even without
extending their work. But this gets laborious: in weight 7 there
are 32 admissible sequences, and the sum for t(2, 1, 1, 1, 1, 1)
has 64 terms. Blümlein, Broadhurst and Vermaseren are all
particle physicists, with plenty of experience with large-scale
calculations, so they probably have helpful ideas on how to
manage the higher weights.
ME Hoffman
Multiple t-Values
Explicit Formulas in Depth 2
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
One place where the theories of MZVs and MtVs are closely
parallel is in depth 2. It is convenient to define
2i − 2
2i − 2
N(a, b, i) =
+
,
a−1
b−1
where pj = 0 if j > p. It is well-known (and implicit in Euler’s
1775 paper) that
(−1)a
a+b
ζ(a, b) =
− 1 ζ(2n + 1)+
2
a
(
P
ζ(a)ζ(b) − ni=2 N(a, b, i)ζ(2i − 1)ζ(2n − 2i + 2), a even,
Pn
i=2 N(a, b, i)ζ(2i − 1)ζ(2n − 2i + 2), otherwise.
if a + b = 2n + 1 and a, b ≥ 2 (No general formula in terms of
single zeta values is known for ζ(a, b) if a + b is even.)
ME Hoffman
Multiple t-Values
Explicit Formulas in Depth 2 cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
For ζ(2n, 1) one has another formula
2n−1
1 X
ζ(2n, 1) = nζ(2n + 1) −
ζ(i)ζ(2n + 1 − i).
2
i=2
Basu 2008 gave the similar formula
1
t(a, b) = − t(2n + 1)+
2
(
P
t(a)t(b) − ni=2 N(a, b, i) t(2i−2)t(2n−2i+2)
, a even,
22i−1 −1
Pn
t(2i−1)t(2n−2i+2)
, otherwise,
i=2 N(a, b, i)
22i−1 −1
for a + b = 2n + 1. Basu actually stipulates that a, b ≥ 2, but
if we take a = 2n, b = 1 and set t(1) = log 2 the formula is
still valid (as follows from Nakamura and Tasaka 2013).
ME Hoffman
Multiple t-Values
Explicit Formulas in Depth 2 cont’d
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
The proofs used by Basu and Nakamura-Tasaka are elementary
but different. I have been able to prove the a = 2n, b = 1 case,
that is,
1
t(2n, 1) = − t(2n + 1) + t(2n) log 2
2
n
X
t(2i − 1)t(2n − 2i + 2)
−
,
22i−1 − 1
i=2
using yet another elementary method. I believe this can be
extended to the general case a + b = 2n + 1, though I haven’t
yet done this. My method involves modified Tornheim sums.
ME Hoffman
Multiple t-Values
Tornheim Sums
Multiple
t-Values
ME Hoffman
The depth 2 formulas for the MZV case can all be rather neatly
proved using sums invented in Tornheim 1950:
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
T (a, b, c) =
∞ X
∞
X
i=1 j=1
1
.
+ j)c
i a j b (i
They have the properties T (a, b, c) = T (b, a, c),
T (a, 0, c) = ζ(c, a) for c > 1, T (a, b, 0) = ζ(a)ζ(b) for
a, b ≥ 2, and
T (a, b, c) = T (a − 1, b, c + 1) + T (a, b − 1, c + 1)
for a, b ≥ 1.
ME Hoffman
Multiple t-Values
Modified Tornheim Sums
Multiple
t-Values
ME Hoffman
For the t-values one can use a modified Tornheim sum
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
M(a, b, c) =
∞ X
∞
X
i=1 j=0
1
(2i)a (2j
+
1)b (2i
+ 2j + 1)c
One has M(0, b, c) = t(c, b) for c > 1,
M(a, b, 0) = 2−a ζ(a)t(b) for a, b ≥ 2 and
M(a, b, c) = M(a − 1, b, c + 1) + M(a, b − 1, c + 1)
for a, b ≥ 1, but the symmetry in the first two arguments is lost
and M(a, 0, c) is not particularly simple. Nevertheless, one can
mimic many of the arguments used for Tornheim sums.
ME Hoffman
Multiple t-Values
Tornheim and Modified Tornheim Sums
Multiple
t-Values
ME Hoffman
Outline
Here is an interesting point of difference between T and M. By
partial fractions one has
Introduction
∞
∞ X
1
1 X 1
−
,
T (1, p, 1) =
j p+1
i
i +j
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
j=1
i=1
and the inner sum telescopes to give
∞
X
1
1
1 + ··· +
= ζ(p + 2) + ζ(p + 1, 1).
j p+1
i
j=1
ME Hoffman
Multiple t-Values
Tornheim and Modified Tornheim Sums cont’d
Multiple
t-Values
ME Hoffman
But in the corresponding sum
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
M(1, p, 1) =
∞
X
j=0
∞ X
1
1
1
−
(2j + 1)p+1
2i
2i + 2j + 1
i=1
the inner sum does not telescope, and instead we have
∞
X
j=0
1
1
1
+
·
·
·
+
−
log
2
+
1
+
(2j + 1)p+1
3
2j + 1
= t(p + 2) + t(p + 1, 1) − t(p + 1) log 2.
ME Hoffman
Multiple t-Values
A Formula for t(2n + 1, 1)
Multiple
t-Values
ME Hoffman
Outline
Introduction
MtVs as
images of
quasisymmetric
functions
Iterated
integrals
Calculations
and
conjectures
Depth 2
results
The argument for t(2n, 1) can be modified slightly to give a
formula for t(2n + 1, 1) (but one must pay the price of
including an alternating MZV):
1
t(2n + 1, 1) = t(2n + 1) log 2 − ζ(2n + 1, 1)+
2
(2n + 1)(1 − 22n+1 ) − 22n+2
t(2n + 2)+
4(22n+2 − 1)
n−1 2n+1
X
2
− 22i+1 − 22n−2i+1 + 1
t(2i + 1)t(2n − 2i + 1).
2(22i+1 − 1)(22n−2i+1 − 1)
i=1
ME Hoffman
Multiple t-Values