
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>...>nm≥11ns11ns22..nsmm
To be convergent s1,s2,sm−1≥2,sm≥1
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 independentDimension
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=∑m∑n1m(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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