On Friday I made a post about multiple zeta values and was frustrated by my inability to find, for example, ζ(2,1).
Sarah Carr has informed me of the missing relation, "Hoffman's relation", which is used for calculating non-convergent ζ values:
For any convergent sequence of positive integers, k = (k1, ... ,kd) and its
corresponding sequence of 0's and 1's, ε = (0k1-11,...,0kd-11), then Σσ ζ(σ) = 0 where σ runs over all the terms in (1)*k - (1) Ш ε
I'm not particularly interested in finding it anymore (I have bigger fish to fry at the moment), but in the interests of completeness I wanted to make sure I had this right.
Showing posts with label Carr. Show all posts
Showing posts with label Carr. Show all posts
17 September 2007
14 September 2007
multiple zeta values
Today I learned about Multiple Zeta Values in my department's graduate student "pizza" seminar. The speaker was Sarah Carr; the link is to her notes. Some of what follows basically rehashes definitions and conjectures from her notes; some of this is my own thoughts. In the future, you might expect to find math that I don't really understand but am pretending to on Fridays, because that's when this seminar meets.
One can define the "multiple zeta value" of a sequence of positive integers (k1, ..., kd), with k1 ≥ 2 (a condition which is needed for convergence), in the following way:
The ordinary Riemann zeta function is just the special case where d=1.
It turns out that these obey certain nice relations which can be found basically by just looking at the sums, for example
ζ(a) ζ(b) = ζ(a, b) + ζ(b, a) + ζ(a+b).
This allows one to compute some of these values; for example, if a = b = 2, we get
ζ(2)2 = 2 ζ(2, 2) + ζ(4)
and using a certain well-known results of Euler, namely that ζ(2) = π2/6 and ζ(4) = π4/90, we get ζ(2,2) = π4/120. Of course, one doesn't want to write ζ over and over again when studying these things, so we'd write something like
(a) * (b) = (a, b) + (b, a) + (a+b)
and this "*" is an example what's called the "stuffle product". (I swear I'm not making this name up!) You can read Carr's notes for the definition in general.
There's a natural way in which we can view sequences of integers as sequences of 0's and 1's; namely, replace each occurence of a by a-1 0's followed by a 1, so, for example, the sequence (2, 3) becomes (0, 1, 0, 0, 1). On these sequences one can define a relation called the "shuffle product", on which one has, for example,
(0, 1) Ш (0, 1) = 2(0, 1, 0, 1) + 4(0, 0, 1, 1)
or, in the original notation,
(2) Ш (2) = 2(2, 2) + 4(3, 1).
Sticking the ζs back in and turning Ш into multiplication is allowed; this is a result of Kontsevich. The proof hinges on a representation of ζ values as integrals, which is pretty natural; the integrals in question have nice power series expansions that coincide with the definition of the ζ values. Doing this, you get
ζ(2)2 = 2ζ(2,2) + 4ζ(3,1)
from which we conclude that ζ(3,1) = ζ(4)/4 = π4/360.
It turns out, though, that the relations one gets from considering the * and Ш operations are graded, in the sense that given a relation among the ζ values, the sum of the arguments in each term of that relation (the "weight" of each term) will be the same. For example, in the relation
ζ(2)2 = 2ζ(2,2) + 4ζ(3,1)
the three terms have weight 2+2, 2+2, and 3+1 respectively. It's conjectured that all the relations among ζ values come from * and Ш, from which it would follow are no relations among ζ values of different weight; this would mean that all ζ values are transcendental. Since putting a sequence which sums to m and one which sums to n into either * or Ш gives a sequence which sums to m+n, this would mean that the ζ values form a graded algebra.
I also don't know how many relations there are among ζ values of the same weight; one might hope that there are enough that we can find all the ζ values of even weight exactly by purely algebraic means, given that we know ζ(2n)? (In particular, this would imply that ζ of any sequence summing to 2n is π2n times some rational number; above, we see that ζ(4), ζ(3,1) and ζ(2,2) are all rational multiples of π4. But I don't have too much hope for that conjecture, because I can't even find ζ(2,1,1) that way! (I think that my inability to do this would follow from a conjecture of Zagier mentioned in Carr's notes, on the dimension of the grade-n part of the algebra of ζ values, but I don't trust myself.)
CORRECTION, Monday, September 17: There's a missing relation that I didn't know about. See this post.
One can define the "multiple zeta value" of a sequence of positive integers (k1, ..., kd), with k1 ≥ 2 (a condition which is needed for convergence), in the following way:
The ordinary Riemann zeta function is just the special case where d=1.
It turns out that these obey certain nice relations which can be found basically by just looking at the sums, for example
ζ(a) ζ(b) = ζ(a, b) + ζ(b, a) + ζ(a+b).
This allows one to compute some of these values; for example, if a = b = 2, we get
ζ(2)2 = 2 ζ(2, 2) + ζ(4)
and using a certain well-known results of Euler, namely that ζ(2) = π2/6 and ζ(4) = π4/90, we get ζ(2,2) = π4/120. Of course, one doesn't want to write ζ over and over again when studying these things, so we'd write something like
(a) * (b) = (a, b) + (b, a) + (a+b)
and this "*" is an example what's called the "stuffle product". (I swear I'm not making this name up!) You can read Carr's notes for the definition in general.
There's a natural way in which we can view sequences of integers as sequences of 0's and 1's; namely, replace each occurence of a by a-1 0's followed by a 1, so, for example, the sequence (2, 3) becomes (0, 1, 0, 0, 1). On these sequences one can define a relation called the "shuffle product", on which one has, for example,
(0, 1) Ш (0, 1) = 2(0, 1, 0, 1) + 4(0, 0, 1, 1)
or, in the original notation,
(2) Ш (2) = 2(2, 2) + 4(3, 1).
Sticking the ζs back in and turning Ш into multiplication is allowed; this is a result of Kontsevich. The proof hinges on a representation of ζ values as integrals, which is pretty natural; the integrals in question have nice power series expansions that coincide with the definition of the ζ values. Doing this, you get
ζ(2)2 = 2ζ(2,2) + 4ζ(3,1)
from which we conclude that ζ(3,1) = ζ(4)/4 = π4/360.
It turns out, though, that the relations one gets from considering the * and Ш operations are graded, in the sense that given a relation among the ζ values, the sum of the arguments in each term of that relation (the "weight" of each term) will be the same. For example, in the relation
ζ(2)2 = 2ζ(2,2) + 4ζ(3,1)
the three terms have weight 2+2, 2+2, and 3+1 respectively. It's conjectured that all the relations among ζ values come from * and Ш, from which it would follow are no relations among ζ values of different weight; this would mean that all ζ values are transcendental. Since putting a sequence which sums to m and one which sums to n into either * or Ш gives a sequence which sums to m+n, this would mean that the ζ values form a graded algebra.
I also don't know how many relations there are among ζ values of the same weight; one might hope that there are enough that we can find all the ζ values of even weight exactly by purely algebraic means, given that we know ζ(2n)? (In particular, this would imply that ζ of any sequence summing to 2n is π2n times some rational number; above, we see that ζ(4), ζ(3,1) and ζ(2,2) are all rational multiples of π4. But I don't have too much hope for that conjecture, because I can't even find ζ(2,1,1) that way! (I think that my inability to do this would follow from a conjecture of Zagier mentioned in Carr's notes, on the dimension of the grade-n part of the algebra of ζ values, but I don't trust myself.)
CORRECTION, Monday, September 17: There's a missing relation that I didn't know about. See this post.
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