Multiplicity Function Of Multiple ζ Function And Rational Point On Hyperelliptic Curve

The multiple zeta functions, are also called multiple harmonic series, or Euler-Zagier sums, which have attracted considerable interest in recent times. The research on such sums is not only important to general zeta function theory, but also touches upon domains such as arithmetic geometry, Galois represen-tations, invariants for knots, quantum groups, etc. The ultimate aim for the researchers in this area is to exhibit all of the identities among multiple zeta functions. In this thesis, we use harmonic shuffle relation to find some identities among these multiple zeta functions.Here we introduce three kinds of the most famous multiple zeta functions:(1)(original) multiple zeta functions(2) alternating multiple zeta functions(3) Hurwitz multiple zeta functions where xi∈[0,1), i=1,2,…,k.In particularly, when the arguments are all positive integers, we refer the above three kinds of multiple zeta functions as corresponding multiple zeta values. In this thesis, we obtain the following results.1. When k=3,4,5, n≥k, by using harmonic shuffle relation, we obtain the identities for the sum of multiple zeta values 2.When k=2,3,4,n≥k,by using harmonic shuffle relaton,we obtain the identities for the sum of alternating multiple zeta values3.When k=2,3,4,5,n≥k,by using harmonic shuffle relation,we obtain the identities for the sum of Hurwitz multiple zeta values and4.We obtain the identities for the series where a(n,k)is the expression of(1),(2)or(3).5.We obtain the congruence of for modular p and p2,respectively,where σi∈{-1,1),i=1,2,3.6. We obtain all rational points(x,y)on the superelliptic curves yk=x(x+2),yk=x(x+2)(x+3),yk=x(x+1)(x+3)and yk=x(x+2m).Meanwhile, we discuss the rational points(x,y)on the superelliptic curves(lx)=yk.