
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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