Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Some Prehistory of Finite Multiple Zeta Values
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
Michael E. Hoffman
A New Start:
QuasiSymmetric
Functions
U. S. Naval Academy
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Kyushu University
Workshop on Multiple Zeta Values
22 August 2014
Michael E. Hoffman
Prehistory of FMZVs
Outline
Prehistory of
FMZVs
Michael E.
Hoffman
1 Multiple Zeta Values
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
2 Multiple Harmonic Sums mod p
3 A New Start: Quasi-Symmetric Functions
4 True Duality at Last
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5 Back to Low Weights
Michael E. Hoffman
Prehistory of FMZVs
Multiple Zeta Values
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
For positive integers i1 , i2 , . . . ik with i1 > 1 (to ensure
convergence), the multiple zeta value (henceforth MZV)
ζ(i1 , i2 , . . . , ik ) is the sum of the k-fold series
X
1
.
ik
i1 i2
n1 >n2 ···>nk ≥1 n1 n2 · · · nk
We call k the depth of ζ(i1 , . . . , ik ), and i1 + · · · + ik its weight.
Of course, the MZVs of depth one are the values ζ(i) of the
Riemann zeta function at positive integers i > 1, for which
Euler proved the formula
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ζ(2i) =
(2π)2i
|B2i |,
2(2i)!
but he also studied depth-two MZVs. In the late 1980’s several
people starting studying MZVs of general depth.
Michael E. Hoffman
Prehistory of FMZVs
My Early Work on MZVs
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
In the summer of 1988, my colleague Courtney Moen interested
me in what he called the “sum conjecture”: for n > k ≥ 2,
X
ζ(a1 , a2 , . . . , ak ) = ζ(n).
a1 >1, a1 +···+ak =n
The case k = 2 had been proved by Euler in his 1776 paper.
Courtney eventually came up with a very long proof of the case
k = 3. I plunged into the search for a general proof, but
instead discovered two other properties of MZVs: the duality
and derivation theorems. Neither gave a quick proof of the sum
conjecture, though the derivation theorem reduced the k = 3
case to a three-line proof.
Michael E. Hoffman
Prehistory of FMZVs
Duality of MZVs
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
Because of the influence of the duality theorem on my later
work, I state it here, in the form I gave it in my 1992 paper. Let
I = (i1 , . . . , ik ) be a composition (finite sequence of positive
integers) with i1 > 1. Let S(I ) = (i1 , i1 + i2 , . . . , i1 + · · · + ik )
be the corresponding sequence of partial sums. If In is the set
of strictly increasing sequences consisting of integers from
{1, 2, . . . , n}, then there are two involutions on In : R given by
R(a1 , . . . , ak ) = (n + 1 − ak , . . . , n + 1 − a1 ),
and C which takes (a1 , . . . , ak ) to its complement in
{1, 2, . . . , n}, arranged in increasing order. Then I defined
τ (I ) = S −1 CRS(I ), and the duality theorem was the result that
ζ(τ (I )) = ζ(I ).
Michael E. Hoffman
Prehistory of FMZVs
Duality of MZVs cont’d
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Actually I couldn’t prove the duality theorem in general, but
only in the “height 1” case where I = (a + 1, 1, . . . , 1) for a
positive integer a. In this case the duality theorem reads
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
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Functions
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ζ(a + 1, 1, . . . , 1) = ζ(b + 1, 1, . . . , 1).
| {z }
| {z }
(1)
a−1
b−1
Of course now we know that the general duality theorem
follows immediately from the iterated-integral representation of
MZVs. But in those early days duality was still pretty
mysterious to me: I had guessed it from doing extensive
calculations in weight ≤ 6.
Michael E. Hoffman
Prehistory of FMZVs
Truncated Multiple Harmonic Sums
Prehistory of
FMZVs
Michael E.
Hoffman
The first phase of my joint work with Courtney on MZVs was
done by about 1992. We then spent some time trying to
understand the truncated multiple harmonic sums (MHSs)
Outline
ζn (i1 , . . . , ik ) =
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
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X
i1 i2
n≥n1 >n2 ···>nk ≥1 n1 n2
1
· · · nkik
which are simply rational numbers. They still had the “stuffle”
multiplication, e.g., ζn (2)ζn (3) = ζn (2, 3) + ζn (3, 2) + ζn (5),
but now any composition could be the exponent string since
convergence wasn’t an issue. In particular,
1 n+1
ζn (1, . . . , 1) =
,
| {z }
n! r + 1
r
where
n k
is the unsigned Stirling number of the first kind.
Michael E. Hoffman
Prehistory of FMZVs
Mod p Finite Sums
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
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Functions
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If p is prime, all the denominators in ζp−1 (I ) are invertible in
Z/pZ, and so ζp−1 (I ) makes sense as an element of that ring.
Courtney and I were already thinking about mod p values of
ζp−1 (I ) when Y. Matiyasevich published a note in the
“Unsolved Problems” section of the American Mathematical
Monthly (January 1992) titled “What divisibility properties to
generalized harmonic sums have?”. It presented several
conjectures about ζp−1 (1, . . . , 1) without any awareness of the
Stirling number formula above. (The May 1992 issue carried a
sort of retraction, giving some references the author should
have been aware of.) Courtney and I had submitted a note
proving the conjectures, but all such submissions were rejected.
Nevertheless, we continued studying the mod p values of the
sums ζp−1 (I ).
Michael E. Hoffman
Prehistory of FMZVs
Wolstenholme’s Theorem
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
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The most obvious result concerning such sums was
Wolstenholme’s theorem that
ζp−1 (1) ≡ 0
mod p
for prime p > 2, and Courtney and I had no trouble
generalizing it to
ζp−1 (r ) ≡ 0
mod p
for prime p > r + 1. So the single zeta values were trivial in
this theory.
Michael E. Hoffman
Prehistory of FMZVs
What Ring Were We Working In?
Prehistory of
FMZVs
Michael E.
Hoffman
A New Start:
QuasiSymmetric
Functions
Let me digress for a a moment to say that early on, we didn’t
worry too much about restrictions on p in our formulas: we
regarded two expressions as the same if they were congruent
mod p for p sufficiently large. We never formalized this by
working in the ring
Q
Z/pZ
Pp prime
p prime Z/pZ
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as is usual nowadays, but it was the same idea.
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
Back to Low
Weights
Michael E. Hoffman
Prehistory of FMZVs
Symmetric Sums
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
In my 1992 paper on MZVs I gave a result expressing
symmetric sums of MZVs as rational polynomials in the single
zeta values, e.g.,
1
1
ζ(3, 2, 2) + ζ(2, 3, 2) + ζ(2, 2, 3) = ζ(3)ζ(2)2 − ζ(3)ζ(4)
2
2
− ζ(2)ζ(5) + ζ(7)
The result carries over to MHSs, and in view of the
Wolstenholme-type results this meant symmetric sums of
MHSs were simply 0 mod p. The first weight where one has an
asymmetrical sum is 3, and ζp−1 (2, 1) is not 0 mod p; in fact
we saw quickly that ζp−1 (2, 1) ≡ Bp−3 mod p for p > 3.
Michael E. Hoffman
Prehistory of FMZVs
Motivation
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
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Weights
This gave us some motivation to pursue the study of mod p
MHSs. We thought of them as a sort of “toy model” for
MZVs. A basic question about MZVs is whether they are really
something beyond single zeta values. It certainly seems that
they are, since (for example) nobody has been able to write
ζ(6, 2) as a rational polynomial in the single zeta values; but
we also don’t know how to prove ζ(6, 2) isn’t such a
polynomial (or for that matter that it isn’t rational!). For
MHSs all the “single zetas” are zero mod p, and yet we had
multiple sums that were clearly nonzero. It was easy to write
programs to produce the mod p values of MHSs, and thus easy
to test conjectures about them.
Michael E. Hoffman
Prehistory of FMZVs
Depths 2 and 3
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
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For depth 2 we had the general result
(−1)i i + j
Bp−i−j
ζp−1 (i, j) ≡
i +j
i
mod p,
(2)
where p > i + j + 1 and Bn is the nth Bernoulli number, and
we had a similar but more complicated result describing
ζp−1 (i, j, k) in terms of binomial coefficients and Bernoulli
numbers. Of course congruence (2) implies ζp−1 (i, j) ≡ 0
mod p if i + j is even, and our depth-3 result implied that
ζp−1 (i, j, k) was congruent mod p to a multiple of
ζp−1 (i + j + k − 1, 1) if i + j + k was odd.
Michael E. Hoffman
Prehistory of FMZVs
My Search for Duality
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
I was eager to find a mod p counterpart of the duality theorem
for MZVs. In fact, I observed early on that
ζp−1 (a, 1, . . . , 1) ≡ ζp−1 (b, 1, . . . , 1)
| {z }
| {z }
b−1
mod p,
(3)
a−1
and, using mod p properties of (both kinds of) Stirling
numbers, was eventually able to prove it. This led me to
believe I was on the verge of getting a general mod p duality
result. I just had to modify my definition of τ so that it
produced (3) instead of (1).
Michael E. Hoffman
Prehistory of FMZVs
A Disappointment
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
Actually this wasn’t hard to do. In the partial-sum step of the
definition of τ I took all but the last term:
S̄(a1 , . . . , ak ) = (a1 , a1 + a2 , . . . , a1 + · · · + ak−1 )
so that the partial sums are in {1, . . . , n − 1} for
n = a1 + · · · + ak . Then I considered the function
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τ̄ (I ) = S̄ −1 CR S̄(I )
which takes (a, 1, . . . , 1) to (b, 1, . . . , 1). But I was in for a
| {z }
| {z }
b−1
a−1
disappointment: when applied to compositions I not of height
1, the congruence ζp−1 (τ̄ (I )) ≡ ζp−1 (I ) mod p simply isn’t
true.
Michael E. Hoffman
Prehistory of FMZVs
A Discovery Delayed
Prehistory of
FMZVs
Michael E.
Hoffman
I had actually missed the real duality result, and wouldn’t find
it for another five years. In retrospect, the problem was that I
was wedded to using the sums
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
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Weights
1
· · · nkak
X
ζp−1 (a1 , . . . , ak ) =
n a1 n a2
p−1≥n1 >n2 >···>nk ≥1 1 2
rather than
?
ζp−1
(a1 , . . . , ak ) =
X
1
.
· · · nkak
n a1 n a2
p−1≥n1 ≥n2 ≥···≥nk ≥1 1 2
This was understandable given my experience with MZVs
(where the only “clean” duality result involves the unstarrred
sums). But for the mod p MHSs, it is the latter kind of sums
that have the nice duality.
Michael E. Hoffman
Prehistory of FMZVs
Low-Weight Calculations
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
Courtney and I kept up our joint work on mod p multiple
harmonic sums through 1995, when I gave a talk about it at a
sectional meeting of the American Mathematical Society in
Greensboro, North Carolina. We did some calculations in low
weights, using our results for double and triple series mod p.
This allowed us to find, correctly, that in weight 3 everything is
a multiple of ζp−1 (2, 1), that everything in weight 4 is zero
mod p, and that everything in weight 5 is a multiple of
ζp−1 (4, 1). But the calculations got bogged down in weight 6
and higher. The problem was that we didn’t know enough
congruences for sums of higher depth.
Michael E. Hoffman
Prehistory of FMZVs
A Shift of Focus
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
In 1996 I shifted my attention back to ordinary MZVs. I spent
most of that year struggling with the algebra of MZV
multiplication, and eventually discovered that the quasi-shuffle
or “stuffle” algebra was essentially the algebra of
quasi-symmetric functions. This was a solo effort, as Courtney
wasn’t as interested in algebra as I was. After I wrote my 1997
paper on MZVs, I broadened my focus somewhat and again
thought about the mod p MHSs, but now from a more
advanced algebraic perspective.
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Michael E. Hoffman
Prehistory of FMZVs
Quasi-Symmetric Functions
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
I started thinking of the algebra of mod p MHSs as just another
homomorphic image of QSym, the algebra of quasi-symmetric
functions. The algebra QSym can be thought of as a
subalgebra of the formal power series ring in x1 , x2 , . . . , where
each xi has degree 1. A formal power series f of bounded
degree is quasi-symmetric if, for any i1 < i2 < · · · < ik , the
coefficient in f of xia11 xia22 · · · xiakk is the same as the coefficient in
f of x1a1 x2a2 · · · xkak . Any quasi-symmetric function is a linear
combination of the monomial quasi-symmetric functions
X
M(a1 ,a2 ,...,ak ) =
xia11 xia22 · · · xiakk ,
i1 <i2 <···<ik
which are indexed by compositions (a1 , . . . , ak ).
Michael E. Hoffman
Prehistory of FMZVs
MZVs, MHSs, and Quasi-Symmetric Functions
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
One can think of MZVs as the images under a homomorphism
ζ : QSym0 → R, where QSym0 is the subalgebra generated by
the M(a1 ,...,ak ) with ak > 1, given by
ζ(M(a1 ,...,ak ) ) = ζ(ak , . . . , a1 );
the homomorphism simply sends xj to 1j . Similarly, for each
prime p there is a homomorphism ζp−1 : QSym → Z/pZ
sending
(
1
mod p, if j < p,
xj → j
0,
otherwise,
so that ζp−1 (M(a1 ,...,ak ) ) = ζp−1 (ak , . . . , a1 ).
Michael E. Hoffman
Prehistory of FMZVs
Multiple Harmonic Sums as Images
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
For a composition I = (i1 , . . . , ik ), we write `(I ) = k (the
length of I ) and |I | = i1 + · · · + ik (the weight of I ). Also, we
write Ī for the reverse (ik , . . . , i1 ) of I . Then our
homomorphism ζp−1 : QSym → Z/pZ sends MI to ζp−1 (Ī ).
? (Ī ) is the image of
Also, ζp−1
X
EI =
MJ ,
(4)
JI
where is the refinement order on compositions, i.e. J I if
J can be obtained by combining adjacent parts of I . Now while
the “fundamental quasi-symmetric functions”
X
FI =
MJ
JI
were well studied, the EI were apparently new.
Michael E. Hoffman
Prehistory of FMZVs
QSym as a Hopf Algebra
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
Actually the EI weren’t entirely absent from the literature on
QSym (though they hadn’t been named). QSym has a Hopf
algebra structure given by
X
∆(MI ) =
MI1 ⊗ MI2 ,
I1 I2 =I
where the sum is over all expressions of I as the juxtaposition
of compositions I1 , I2 . Then the antipode S of the Hopf algebra
QSym is given by
S(MI ) = (−1)`(I ) EĪ .
Michael E. Hoffman
Prehistory of FMZVs
(5)
QSym as a Hopf Algebra cont’d
Prehistory of
FMZVs
Michael E.
Hoffman
Since QSym is a commutative Hopf algebra, S is an
automorphism of QSym as an algebra, and S 2 = id. Thus any
formula in the MI , e.g.,
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
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M(2) M(1,3) = M(2,1,3) + M(1,2,3) + M(1,3,2) + M(3,3) + M(1,5)
has a corresponding version in the EI by applying S:
−E(2) E(3,1) = −E(3,1,2) − E(3,2,1) − E(2,3,1) + E(3,3) + E(5,1) .
There was no reason to think of the MI as more fundamental
than the EI , and by Möbius inversion of equation (4)
X
MI =
(−1)`(I )−`(J) EJ .
JI
Michael E. Hoffman
Prehistory of FMZVs
QSym as a Hopf Algebra cont’d
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Also, by comparing (5) with another formula for S one has
X
(6)
(−1)`(I ) EI =
(−1)k MI1 · · · MIk ,
I1 I2 ···Ik =Ī
where the sum is over all expressions of Ī as a juxtaposition of
compositions. For example,
− E(2,1,3) = −M(3,1,2) + M(3) M(1,2) + M(3,1) M(2)
− M(3) M(1) M(2) .
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Michael E. Hoffman
Prehistory of FMZVs
Operators on Sequences
Prehistory of
FMZVs
Michael E.
Hoffman
I also played around with operators on sequences {a(n)}∞
n=0 . In
particular, one has the partial-sum operator Σ given by
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
Σa(n) =
a(n)
i=0
and the operator ∇ given by
∇a(n) =
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n
X
n
X
i=0
(−1)i
n
a(i).
i
It is easy to see that ∇ and Σ generate an infinite dihedral
group under composition, i.e, ∇2 = id and ∇Σ = Σ−1 ∇.
Michael E. Hoffman
Prehistory of FMZVs
True Duality at Last
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
If for each composition I we define a sequence ζI? by
ζI? (n) = ζn? (I ), then I noticed that
Σ∇ζI? = −ζI?∗ ,
where I ∗ = S̄ −1 C S̄(I ) in the notation introduced above (so
(3, 1, 2)∗ = (1, 1, 3, 1)). Also, it is easy to see that
Σ∇a(p) ≡ a(p) mod p for prime p. So now I had my true
duality theorem:
?
?
ζp−1
(I ) ≡ −ζp−1
(I ∗ )
mod p.
I don’t know exactly when I proved this, but it was prior to
September 2000 (when I gave a talk about it).
Michael E. Hoffman
Prehistory of FMZVs
True and False Duality
Prehistory of
FMZVs
Michael E.
Hoffman
It is interesting to see how I was led astray by the result (3),
which is essentially a coincidence. It is a general (and easily
proved) fact that
Outline
Multiple Zeta
Values
?
?
ζp−1
(ik , . . . , i1 ) ≡ (−1)i1 +···+ik ζp−1
(i1 , . . . , ik )
mod p.
(7)
Multiple
Harmonic
Sums mod p
Since (a, 1, . . . , 1)∗ = (1, . . . , 1, b), if we apply the duality
| {z }
| {z }
A New Start:
QuasiSymmetric
Functions
theorem to I = (a, 1, . . . , 1) and then use the congruence (7),
| {z }
True Duality
at Last
we have
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Weights
a−1
b−1
b−1
?
?
(−1)a ζp−1
(a, 1, . . . , 1) = (−1)b ζp−1
(b, 1, . . . , 1)
| {z }
| {z }
a−1
b−1
Michael E. Hoffman
Prehistory of FMZVs
True and False Duality cont’d
Prehistory of
FMZVs
Michael E.
Hoffman
Now applying ζp−1 to equation (6) gives
X
?
(−1)`(I ) ζp−1
(I ) =
(−1)k ζp−1 (I1 ) · · · ζp−1 (Ik ).
I1 I2 ···Ik =Ī
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
(8)
If I = (a, 1, . . . , 1), then mod p the right-hand side collapses to
| {z }
b−1
a single term (since ζp−1 (1, . . . 1) ≡ 0 mod p) and we have
?
(a, 1, . . . , 1)
ζp−1 (a, 1, . . . , 1) ≡ (−1)a ζp−1
| {z }
| {z }
b−1
mod p,
(9)
b−1
from which congruence (3) follows. But congruence (9)
depends very strongly on the special features of the
composition (a, 1, . . . , 1). For I not of this form the right-hand
side of (8) can have many terms that are nonzero mod p.
Michael E. Hoffman
Prehistory of FMZVs
A Useful Byproduct
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Nevertheless, my pursuit of “false duality” left me with one
byproduct that was of some use later. As a result of my
investigation of the MHSs ζn (a, 1, . . . , 1), I proved that
| {z }
b−1
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
?
(−1)a ζp−1
(a, 1, . . . , 1) ≡ ζp−1 (a, 1, . . . , 1) ≡
| {z }
| {z }
b−1
p−b
X
b−1
j
(−1) (−j)
j=1
p−a
p−b
(j − 1)!
j
mod p,
where kn is the Stirling number of the second kind. This
allowed me to compute these MHSs much more effectively than
doing it directly from the definition.
Michael E. Hoffman
Prehistory of FMZVs
A Barrier Removed
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
Now that I had a valid duality result, the barrier to computing
MHSs in low weights was removed. For fixed weight,
small-depth MHSs are dual to large-depth MHSs because of
the relation `(I ) + `(I ∗ ) = |I | + 1. Nevertheless, though I
included a section on finite sums mod p in my survey article
published in 2005 (but written two years earlier), I didn’t really
have the opportunity to renew my computational attack on low
weights until my sabbatical in 2003-4. I made use of both the
duality theorem and some results derived from equation (6)
? (I ) in terms of ζ ? (J) for J of smaller
that expressed ζp−1
p−1
depth. At that time I worked out generators through weight 9.
The results were as follows.
Michael E. Hoffman
Prehistory of FMZVs
Results for |I | ≤ 9
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
|I | ≤ 2: all zero
? (2, 1) ≡ B
|I | = 3: mod p all MHSs multiples of ζp−1
p−3
mod p.
|I | = 4: all zero
? (4, 1) ≡ B
|I | = 5: mod p all MHSs multiples of ζp−1
p−5
mod p.
? (4, 1, 1) ≡ − 1 B 2
|I | = 6: mod p all MHSs multiples of ζp−1
6 p−3
? (6, 1) ≡ B
|I | = 7: mod p all MHSs combinations of ζp−1
p−7
? (4, 1, 1).
mod p and ζp−1
? (6, 1, 1) and
|I | = 8: mod p all MHSs combinations of ζp−1
?
?
ζp−1 (2, 1)ζp−1 (4, 1) ≡ Bp−3 Bp−5 mod p.
? (6, 1, 1, 1),
|I | = 9: mod p all MHSs combinations of ζp−1
?
ζp−1 (8, 1) ≡ Bp−9 mod p, and
? (2, 1)ζ ? (4, 1, 1) ≡ − 1 B 3
ζp−1
p−1
6 p−3 mod p.
Michael E. Hoffman
Prehistory of FMZVs
Missing Congruences?
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
This gives counts of
wt.
gens.
1
0
2
0
3
1
4
0
5
1
6
1
7
2
8
2
9
3
In weights 7,8,9 I had the nagging suspicion that I might have
missed some congruences. The only way I could think of to
test independence was to find a prime that divided one
Bernoulli number. In weight 7 this failed because for no known
primes p does p|Bp−7 . In weight 8 I was reassured because for
? (4, 1) ≡ 0 mod p but ζ ? (6, 1, 1) 6≡ 0 mod p. In
p = 37, ζp−1
p−1
weight 9 I looked at the only known p’s for which
? (8, 1) ≡ 0 mod p, namely p = 67 and p = 877, and at
ζp−1
p = 16, 843, for which p|Bp−3 . This gave inconclusive results.
Michael E. Hoffman
Prehistory of FMZVs
Missing Congruences? cont’d
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
Later I learned that Jianqing Zhao had done similar
computations, and he informed me that he suspected a
congruence
1 ?
?
?
(2, 1)ζp−1
(4, 1, 1)
ζp−1
(6, 1, 1, 1) ≡ − ζp−1
9
1889 ?
+
ζ (8, 1) mod p (10)
648 p−1
in weight 9. When I saw that the congruence (10) was
consistent with the values I’d computed at the three primes
p = 67, p = 877, and p = 16, 843, I was strongly inclined to
believe it.
Michael E. Hoffman
Prehistory of FMZVs
Recent Developments
Prehistory of
FMZVs
Michael E.
Hoffman
Outline
Multiple Zeta
Values
Multiple
Harmonic
Sums mod p
A New Start:
QuasiSymmetric
Functions
True Duality
at Last
Back to Low
Weights
In June 2012 I saw the preprint “New properties of multiple
harmonic series modulo p and p-analogues of Leschinger’s
series,” by Kh. Hessami Pilehrood, T. Hessami Pilehrood and
R. Tauraso; it has since appeared in the AMS Transactions.
The authors not only proved the congruence Zhao suspected,
but also found another congruence
?
ζp−1
(4, 1, 1, 1) ≡
27 ?
ζ (6, 1)
16 p−1
mod p
in weight 7. With these results, the revised counts are
wt.
gens.
1
0
2
0
3
1
4
0
5
1
6
1
7
1
8
2
9
2
and satisfy the Padovan recurrence Pn = Pn−2 + Pn−3 .
Michael E. Hoffman
Prehistory of FMZVs