Zeta-functions Of Root Systems Research Articles

On $q$-analogues of Zeta Functions of Root Systems II

In a previous paper, we introduced functions $\zeta_r(s,a,\Delta;q)$, which are $q$-analogues of zeta functions associated with root systems $\Delta$, and their $p$-deformations $\zeta_r(s,a,\beta,\Delta;p,q)$. It is known that zeta functions of root systems satisfy certain functional relations including Witten's volume formula. In this paper, we consider $q$-extensions of these functional relations for $\Delta=\Delta(A_2), \ \Delta(A_3), \ \Delta(B_2), \ \Delta(G_2)$. We also generalize expressions of ``Weyl group symmetric'' linear combination of $\zeta_r(s,a,\beta,\Delta;p,q)$ for $\Delta=\Delta(A_1), \ \Delta(A_2), \ \Delta(A_3)$, which were obtained in the previous paper, to arbitrary root systems $\Delta$ by using an identity due to Macdonald.

Multiple zeta values and zeta-functions of root systems

We propose the viewpoint that the $r$-ple zeta-function of Euler-Zagier type can be regarded as a specialization of the zeta-function associated with the root system of $C_{r}$ type. From this viewpoint, we can see that Zagier’s well-known formula for multiple zeta values (MZVs) coincides with Witten’s volume formula associated with a sub-root system of $C_{r}$ type. Based on this observation, we generalize Zagier’s formula and also give analogous results which correspond to a sub-root system of $B_{r}$ type. We announce those results as well as some relevant results for partial multiple zeta values.

On Witten multiple zeta-functions associated with semisimple Lie algebras II

We prove asymptotic formulas for mean square values of the Euler double zeta-function $\zeta_2(s_0,s)$, with respect to $\Im s$. Those formulas enable us to propose a double analogue of the Lindel{\"o}f hypothesis.