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SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sequential Zeta Values
Michael E. Hoffman
U. S. Naval Academy
Number Theory TalkMax-Planck-Institut für Mathematik, Bonn
17 June 2015
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Outline
1 Introduction
2 Proof of the Sum Theorem for H-series
3 Iterated Integrals
4 Sequential Zeta Values
5 Sum Conjecture for SZVs
6 Product Structure
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Introduction
For positive integers a1, . . . , ak with a1 > 1 we define thecorresponding multiple zeta value (MZV) by
ζ(a1, a2, . . . , ak) =∑
n1>n2>···>nk≥1
1
na11 na22 · · · n
akk
. (1)
One calls k the depth and a1 + · · ·+ ak the weight. Euleralready studied the cases of depth 1 and depth 2, but arguablythe present era of MZVs of general depth began with the proofof the “sum theorem”∑
a1+···+ak=n, a1>1, ai≥1ζ(a1, . . . , ak) = ζ(n). (2)
This was proved by Euler for depth 2, by C. Moen for depth 3,and by A. Granville and D. Zagier for general depth.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Introduction cont’d
Recently Moen and I proved a rather different sum theorem.Define, for nonnegative integers a1, . . . , ak witha1 + · · ·+ ak ≥ 2, the series
ζ(a1|a2| · · · |ak) =∞∑n=1
1
(n + k − 1)a1(n + k − 2)a2 · · · nak. (3)
(In our paper these are called “H-series”). Note that this is asingle sum, in contrast to the k-fold sum (1). Then our result(Integers, 2014) is∑
a1+···+ak=n,ai≥0ζ(a1|a2| · · · |ak) = kζ(n) (4)
for n ≥ 2.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Introduction cont’d
The H-series (3), unlike MZVs, don’t have the property thatthe product of two such series is a finite sum of series of thesame kind. After some experimentation I arrived at a definitionof “sequential zeta values” (SZVs) which has the property thatthe product of two SZVs is a finite sum of SZVs. Further, thereis a plausible “sum conjecture” for SZVs that includes thetheorems (2) and (4) as special cases. This gives one hope thatSVZs are interesting mathematical objects. But so far we don’thave plausible number-theoretic or geometric interpretations forSZVs (though most can be expressed as iterated integrals).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series
We begin with a look at the proof of the sum theorem (4) asgiven in our paper; it relies on a chain of lemmas about theH-series. The first, which is entirely trivial, is that
m−1∑k=1
ζ(a1| · · · |ai−1|k |ai+1| · · · |aj−1|m − k |aj+1| · · · |an) =
1
j − i[ζ(a1| · · · |ai−1|0|ai+1| · · · |aj−1|m − 1|aj+1| · · · |an)
− ζ(a1| · · · |ai−1|m − 1|ai+1| · · · |aj−1|0|aj+1| · · · |an)]
for 1 ≤ i < j ≤ n.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series cont’d
From this it follows that∑ai0+···+aik=m
ζ(a1|a2| · · · |an) =
k∑j=1
(−1)j−1H(m−k)i0,ij−1(ij − i0) · · · (ij − ij−1)(ij+1 − ij) · · · (ik − ij)
(5)
for any fixed sequence 1 ≤ i0 < i1 < · · · < ik ≤ n, where
H(m)p,q =
q∑j=p
1
jm.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series cont’d
From the formula (5) it follows that the sum C (k, n;m) of allζ(a1|a2| · · · |an) with with exactly k + 1 of the ai nonzero anda1 + · · ·+ an = m can be written in the form
n−1∑j=1
c(n)k,j
jm−k
for c(n)k,j ∈ Q. It is an easy observation that the rational
numbers c(n)k,j have the symmetry/antisymmetry property
c(n)k,n−j = (−1)
k−1c(n)k,j .
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Lemmas about H-series cont’d
It is also easy to show from equation (5) and thesymmetry/antisymmetry property that
c(n)k,1 = (−1)
k−1c(n)k,n−1 =
1
(n − 1)!
[n
k + 1
],
where[nk
]is the Stirling number of the first kind, i.e., the
number of permutations of {1, 2, . . . , n} with exactly k disjointcycles. The last (rather tricky) lemma is that
c(n)k,j =
j∑q=1
q∑p=1
(−1)p−1[qp
][n+1−qk+2−p
](q − 1)!(n − q)!
for 1 ≤ k, j ≤ n − 1.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Proof of the Sum Theorem for H-series
Now
∑a1+···+an=m, ai≥0
ζ(a1|a2| · · · |an) =n−1∑k=0
C (k, n;m);
recall C (k , n;m) is the sum of those terms with nonzero entriesin exactly k + 1 positions. We have
C (0, n;m) = ζ(0| · · · |0|m)+ζ(0| · · · |m|0)+· · ·+ζ(m|0| · · · |0)
= ζ(m) + ζ(m)− 1 + · · ·+ ζ(m)− 1− 12m− · · · − 1
(n − 1)m
= nζ(m)−n−1∑j=1
n − jjm
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Proof of the Sum Theorem for H-series cont’d
That the remaining terms C (k, n;m) cancel the negative termsabove, i.e., that
n−1∑k=1
C (k , n;m) =n−1∑k=1
n−1∑j=1
c(n)k,j
jm−k=
n−1∑j=1
n − jjm
,
follows from the formula expressing c(n)k,j in terms of Stirling
numbers. It follows that∑a1+···+an=m, ai≥0
ζ(a1|a2| · · · |an) = nζ(m).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Iterated Integral Representation
For the multiple zeta value ζ(a1, . . . , ak) we have thewell-known iterated integral representation
ζ(a1, a2, . . . , ak) =
∫ 10ωa1−10 ω1ω
a2−10 ω1 · · ·ω
ak−10 ω1
where
ω0 =dt
t, ω1 =
dt
1− t.
Provided all the ai are positive, there is a similar representationfor the H-series ζ(a1| · · · |ak), i.e.,
ζ(a1|a2| · · · |ak) =∫ 10ωa1−10 dtω
a2−10 dt · · ·ω
ak−10 ω1.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Iterated Integrals cont’d
We can also represent H-series in the which the sequence haszeroes in positions other than the first or last as iteratedintegrals, e.g.,
ζ(2|0|1) =∫ 10ω0tdtω1
ζ(1|0|0|1|1) =∫ 10
t2dtdtω1
ζ(1|0|2|0|0|2) =∫ 10
tdtω0t2dtω0ω1.
(Initial zeroes don’t affect the value of the series, but trailingzeroes do.)
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Definition of SZVs
The H-series do not form a ring. But there is a larger class ofseries that does form a ring, and includes both MZVs andH-series as special cases. We call these sequential zeta values.Let J1, J2, . . . , Jk be sequences of nonnegative integers, say
Ji = (ai ,1|ai ,2| · · · |ai ,li )
and set `(Ji ) = li , |Ji | = ai ,1 + · · ·+ ai ,li . Then the sequentialzeta value ζ(J1, J2, . . . , Jk) is∑
n1>l1n2>l2 ···>lk−1nk>lk 0
1∏ki=1
∏ai,jj=1(n − j + 1)ai,j
where a >k b means a− b ≥ k . We call |J| =∑k
i=1 |Ji | theweight of the SZV, and (l1, l2, . . . , lk) its shape.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Convergence of SZVs
It is immediate from the definition that
ζ(J1, J2 . . . , Jk) ≤ ζ(|J1|, |J2|, . . . , |Jk |).
Hence ζ(J1, J2, . . . , Jk) converges provided |J1| > 1 and|Ji | ≥ 1 for i ≥ 2. In fact, the following is true.
Proposition
The SZV ζ(J1, J2, . . . , Jk) converges provided |J1| > 1, |Ji | ≥ 1for 1 < i < k, and |J1|+ |J2|+ · · ·+ |Jk | > k.
The only case requiring examination is if |Jk | = 0. In this caseζ(J1, . . . , Jk) can be written∑
n1>l1 ···nk−1>lk−1>lk
nk−1 − lk∏k−1i=1 n
ai,1i (ni − 1)ai,2 · · · (ni − li + 1)ai ,li
,
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Convergence of SZVs cont’d
which is
ζ(J1, . . . , ak−1,1| · · · |ak−1,lk−1 − 1)− ζ(J1, . . . , Jk−1).
This is a sum of convergent SZVs if |Jk−1| > 1. Otherwise thefirst term has last sequence 0, and we can iterate. The onlycase that would produce a non-convergent SZV would be|J1| = 2, |J2| = · · · = |Jk−1| = 1, and |Jk | = 0, but this isexcluded by the hypothesis.We note that a leading 0 in J1 can simply be omitted withoutaffecting the value of the series; so henceforth we assume thatJ1 starts with a nonzero integer. If also |J1| > 1, Ji has notrailing 0 for i < k, |Ji | ≥ 1 for 1 < i < k , and|J1|+ · · ·+ |Jk | > k , we call (J1, . . . , Jk) admissible.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Properties of SZVs
SZVs have some properties from partial fractions. Since
1
(n − p)(n − q)=
1
q − p
(1
n − p− 1
n − q
)it follows that
ζ(J1, . . . , ai ,1| · · · |ai ,q| · · · |ai ,p| · · · |ai ,li , . . . , Jk) =1
q − pζ(J1, . . . , ai ,1| · · · |ai ,p − 1| · · · |ai ,q| · · · |ai ,li , . . . , Jk)
− 1q − p
ζ(J1, . . . , ai ,1| · · · |ai ,p| · · · |ai ,q − 1| · · · |ai ,li , . . . , Jk)
whenever ai ,p, ai ,q > 0.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Properties of SZVs cont’d
Recall that we assume J1 has no leading 0. A trailing 0 in Jk israther complicated to describe in general; one has, e.g.,
ζ(2, 1, 1|0) =∑
i>j>k≥2
1
i2jk=
∑i>j>k≥1
1
i2jk−∑i>j≥2
1
i2j=
ζ(2, 1, 1)−∑i>j≥1
1
i2j+∑i≥2
1
i2= ζ(2, 1, 1)− ζ(2, 1) + ζ(2)− 1.
All other leading and trailing zeroes can be disposed of via
ζ(J1, . . . , Ji |0, Ji+1, . . . , Jk) = ζ(J1, . . . , Ji , 0|Ji+1, . . . , Jk) =ζ(J1, . . . , Ji , Ji+1, . . . , Jk)− ζ(J1, . . . , Ji |Ji+1, . . . , Jk).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Strings of 1’s
By the preceding results we have for p > 1
ζ(1| · · · |1︸ ︷︷ ︸p
, J2, . . . , Jk) =1
p − 1ζ(1| · · · |1︸ ︷︷ ︸
p−1
|J2, J3, . . . , Jk).
Using this one can easily give a formula for ζ(J1, . . . , Jk) wheneach Ji is a string of 1’s:
1
(|J1| − 1)(|J1|+ |J2| − 2) · · · (|J1|+ · · ·+ |Jk | − k)
× 1(|J1|+ · · ·+ |Jk | − k)!
.
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Other Strings
A similar result is
ζ(1| 0| · · · |0︸ ︷︷ ︸p
|1, J2, . . . , Jk) =
1
p + 1
p∑j=0
ζ(1| 0| · · · |0︸ ︷︷ ︸j
|J2, J3, . . . , Jk).
For example,
ζ(1|0|0|1, 2, 1) = 13
[ζ(1|2, 1) + ζ(1|0|2, 1) + ζ(1|0|0|2, 1)] .
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Iterated Integrals Again
Provided that the last entry of Jk is nonzero, the SZVζ(J1, . . . , Jk) can be represented as an iterated integral. Forexample,
ζ(3|1, 2) =∑i ,j≥1
1
(i + j + 1)3(i + j)j2=
∫ 10ω20dtω1ω0ω1
ζ(2, 1|0|1) =∑i ,j≥1
1
(i + j + 2)2(i + 2)i=
∫ 10ω0ω1tdtω1
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sum Conjecture
An MZV is a sequential zeta value of shape (1, 1, . . . , 1), whilean H-series is a sequential zeta value of shape (k). It turns outthat the sum theorems (2) and (4) can be put into a commonform for SZVs. Recall that a sequence (J1, . . . , Jk) isadmissible if J1 does not start with 0, |J1| > 1, Ji has notrailing zero if i < k , |Ji | ≥ 1 for 1 < i < k, and|J1|+ |J2|+ · · ·+ |Jk | > k . For a given sequence S , let Am(S)be the set of admissible sequences of shape S and weight m;the length `(S) of a shape S = (l1, . . . , lk) is k .
Conjecture
For m > `(S), ∑p∈Am(S)
ζ(p) = ζ(m − `(S) + 1).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sum Conjecture cont’d
It is simple to put the sum theorem (4) for H-series into thisform: in the identity∑
a1+···+ak=n, ai≥0ζ(a1| · · · |ak) = kζ(n),
the inadmissible terms on the right-hand side are those witha1 = 0. These add up to∑
a2+···+ak=n, ai≥0ζ(0|a2| · · · |ak) = (k − 1)ζ(n)
since ζ(0|a2| · · · |ak) = ζ(a2| · · · |ak); hence∑a1+···+ak=n, a1≥1, ai≥0
ζ(a1| . . . |ak) = ζ(n).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Sum Conjecture cont’d
To put the MZV sum theorem (2) in this form is a little moreinvolved; to avoid notational complications we treat the specialcase depth 3, weight 5. The MZV sum theorem says
ζ(3, 1, 1) + ζ(2, 1, 2) + ζ(2, 2, 1) = ζ(5) (6)
There are additional admissible triple sums
ζ(3, 2, 0) = ζ(3, 1)− ζ(3, 2)ζ(2, 3, 0) = ζ(2, 2)− ζ(2, 3)ζ(4, 1, 0) = ζ(3)− ζ(4)− ζ(4, 1).
When these are added to the left-hand side of equation (6), theresult is
ζ(3, 1, 1) + ζ(2, 1, 2) + ζ(2, 2, 1) + ζ(3, 1) + ζ(2, 2)
− ζ(3, 2)− ζ(2, 3)− ζ(4, 1)− ζ(4) + ζ(3) = ζ(5− 3 + 1).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Known Cases of Sum Conjecture
So the sum conjecture holds for shape (k) and for shape(1, . . . , 1). So far we have only been able to prove that the sumconjecture holds for a few other shapes. These include allshapes of the form (k ,m) for k ≥ 2. Shapes of the form (1,m)seem to be harder; the conjecture has been proved for shapes(1, 2) and (1, 3). We give the proof for shape (1, 2).
Proposition
If m ≥ 3, then ∑p∈Am(1,2)
ζ(p) = ζ(m − 1).
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Known Cases of Sum Conjecture cont’d
The left-hand side is
ζ(m, 0|0) +∑
i+j=m, i≥2, j≥1[ζ(i , j |0) + ζ(i , 0|j)]
+m−2∑i=2
m−1−i∑j=1
ζ(i , j |m − i − j). (7)
Now
ζ(m, 0|0) = ζ(m − 1)− 2ζ(m)− 1ζ(i , j |0) + ζ(i , 0|j) = 2ζ(i , j)− ζ(i |j)− ζ(i) + 1
so that the first term and first sum of (7) give
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Known Cases of Sum Conjecture cont’d
ζ(m−1)−2ζ(m)−1+m−1∑i=2
[2ζ(i ,m− i)−ζ(i |m− i)−ζ(i)+1]
= ζ(m − 1) + m − 2−m−2∑i=2
ζ(i)− ζ(1|m − 2)
using the sum theorem of MZVs. The second sum in (7) gives
m−2∑i=2
[ζ(i)− ζ(i |m − i − 1) + ζ(i)− 1] =
m−2∑i=2
ζ(i)− ζ(1|m − 3) + ζ(m − 2)− (m − 2)
and the conclusion follows.ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Product of SZVs
The product of two SZVs is a finite sum of SZVs, but theproduct structure (unlike that for MZVs) does not fall into theframework of quasi-shuffle products. As with the product ofMZVs, all shuffles appear in the product, together with termsin which sequences combine; a new feature is that a singlesequence in one factor can “paste together” two or moresequences in the other. Here are some examples.
ζ(2)ζ(2|1) = ζ(2, 2|1) + ζ(2|1, 2) + ζ(4|1) + ζ(2|3)ζ(2, 1)ζ(2|1) = ζ(2, 1, 2|1) + ζ(2, 2|1, 1) + ζ(2|1, 2, 1)
+ζ(2, 3|1) + ζ(2, 2|2) + ζ(4|1, 1) + ζ(2|3, 1) + ζ(4|2)ζ(2|1)2 = 2ζ(2|1, 2|1) + 2ζ(2|3|1) + ζ(4|2)
ME Hoffman Sequential Zeta Values
SequentialZeta Values
ME Hoffman
Outline
Introduction
Proof of theSum Theoremfor H-series
IteratedIntegrals
SequentialZeta Values
SumConjecture forSZVs
ProductStructure
Product Structure cont’d
Here is another example.
ζ(1|1)ζ(2|1) = ζ(2|1, 1|1) + ζ(1|1, 2|1) + ζ(1|3|1)+ ζ(2|2|1) + ζ(3|2).
Since ζ(1|1) = 1, the left-hand side is simply ζ(2|1) = 2− ζ(2).That right-hand side equals this follows from earlier results:
ζ(2|1, 1|1) = ζ(1|1|1) + ζ(2|1)− ζ(2) + 1 + ζ(2|1|1)ζ(1|1, 2|1) = ζ(2|1)− ζ(1|1|1)ζ(1|3|1) = ζ(3|1)− ζ(1|2|1)ζ(2|2|1) = ζ(1|2|1)− ζ(2|1|1)ζ(3|2) = ζ(2)− 1− ζ(2|1)− ζ(3|1)
ME Hoffman Sequential Zeta Values
IntroductionProof of the Sum Theorem for H-seriesIterated IntegralsSequential Zeta ValuesSum Conjecture for SZVsProduct Structure