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Monday, June 18, 2018

Multiple zeta function (Values) Mzv

Multiple zeta Values (Mzv) intro Multiple zeta function (Values) Mzv

Abstract

I just think that multiple zeta function is interesting topic and I just saw the Don Zagier's Video and other videos discussing it. I want to build my note for this interesting series. Hope I can understand motives, periods one day.

Definition:

ζ(s1,s2,..sm)=n1>n2>...>nm11ns11ns22..nsmm
To be convergent s1,s2,sm12,sm1
This is famous generalization for riemann zeta function: ζ(s)=n=11ns
If you are interested in this. You can see my another blog Note for Riemann Hypothesis The famous number traced back to Euler is: ζ(2)=n=11n2=π26
And Euler also proof that k is positive integer: ζ(2k)=n=11n2k=Rational Numbers π2k
I will give a proof in this blog. But up to now, no one can give a closed form for zeta function with odd integers which led to the following conjecture:

Conjecture 1

ζ(3),ζ(5),,,,π2k are algebraic independent

Dimension

There are many relations for mzv. We can define dimension for weight k is a vector space spanned by Q coefficient: Qζ(s1,s2,..sm):s1+s2+...sm=k
. So what is the dimension for this vector space?

Weight=3 --- dimension=1

It seems like we have two dimensional space since we have ζ(3),ζ(2,1)
But actually they are the same: ζ(3)=ζ(2,1)
. proof: Write mn1mn(m+n)=mn1m(m+n)2+mn1n(m+n)2
We can see (after some mediation) mn1m(m+n)2=n>m=11m(n)2=ζ(2,1)
So we have: mn1mn(m+n)=2ζ(2,1)
On the other hand: mn1m(m+n)2=mn1m(1m(m+n))
Use m1m(m+n)=1n(1+12+....1n)=1+14(1+12)+19(1+12+13)+...
=ζ(2,1)+ζ(3)
So we have ζ(3)=ζ(2,1)
.

Nielsen Reflexion Formula

ζ(a)ζ(b)=ζ(a,b)+ζ(b,a)+ζ(a+b)
proof: ζ(a)ζ(b)=n,m1namb=n>m>=11namb+m>n>=11namb+n=m>=11nanb=ζ(a,b)+ζ(b,a)+ζ(a+b)












Continuing:

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