The Incremental Value Of The Generalized Zeta Function Is Approximately Estimated

In the past decade, several kinds of multiple zeta functions have been related to variety subjects such as knot theory, cohomology and even quantum physics. They have attracted increasing attention, especially the studies of reduction and evaluation of the multiple zeta functions. Even though the theory of multiple zeta function develops very fast, they are independent relatively due to their applications to different subjects. In this thesis, we provide a powerful combinatoricalmethod to find some relations among these multiple zeta functions to help us to understand them deeper and better.Here we introduce three kinds of the most famous multiple zeta functions: (1) (original) multiple zeta functions (with depth k and weight s₁+ s₂ +…+s_k):(2) Mordell-Tornheim zeta functions (with depth k and weight s₁+s₂+…+s_k + s):(3) Witten zeta functions:where s∈C, g is a semisimple Lie algebra and p runs over all finite dimensional irreducible representations of g.Especially, 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. We prove that every Mordell-Tornheim zeta value of depth r and weight w can be expressed as a rational linear combination of multiple zeta values of depth r and weight w, then use the reduction theory of multiple zeta values, we prove that every Mordell-Tornheim zeta value of depth at least two and with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth.2.When depth is 2, we define the signed q-Tornheim series, and show an explicit evaluation formula for these series with the weights odd. As a direct corollary, we partially answer an open problem posted in 1985 by M.V. Subbarao and R. Sitaramachandra Rao.3.When depth is 2, we show a very simple and elementary proof of Takashi Nakamura's functional relation forζ_mt,3(a, b, s)+(-1)⁶ζ_mt,3(b,s,a)+(-1)^aζ₍MT,3_{s, a,b}.4.We consider the L-analog of Tornheim series and prove that a more general class of L-analog of Tornheim series can be evaluated.5.We point out that several classes of multiple Witten zeta functions can be reduced or evaluated by multiple zeta functions, we also show a new proof for the formula ofζ_sl(4)(2m).6.We consider the partial sums of multiple zeta values, and prove two new congruences.