Functional relations and universality for several types of multiple zeta functions

Firstly, we prove a functional relation for the Tornheim double zeta function. Using this functional relation, we obtain simple proofs of some known formulas for special values of Tornheim and Euler-Zagier double zeta functions. Secondly, we obtain functional relations for Witten zeta functions by using a double L-values relation. By these functional relations, we obtain new proofs of known results on the Tornheim double zeta function, the Euler-Zagier double zeta function, their alternating and character analogues. Thirdly, we define λ-joint, a′-joint, (λ, λ)-joint, (λ, a′)-joint and (a′, a′)-joint t-universality of Lerch zeta functions and consider the relations among those. Next we show the existence of (λ, λ)joint t-universality. We also show the existence of λ-joint, a′-joint, (λ, a′)-joint and (a′, a′)-joint t-universality by using inversion formulas. Fourthly, we show the following theorems. Suppose 0 < al < 1 are algebraically independent numbers and 0 < λl ≤ 1 for 1 ≤ l ≤ m. Then we have the joint t-universality for Lerch zeta functions L(λl, al, s) for 1 ≤ l ≤ m. Next we generalize Lerch zeta functions, and obtain the joint t-universality for them. In addition, we show examples of the non-existence of the joint t-universality for Lerch zeta functions and generalized Lerch zeta functions.