In this paper, we determine all orbits of the prehomogeneous vector space

(GL₅ × GL_4, ∧² Aff⁵ ⊗ Aff⁴)

rationally over an arbitrary perfect field whose characteristic is not two.

In this paper we first establish several integral identities involving the multiple polylogarithm functions and the Kaneko-Tsumura A-function, which can be thought as a single-variable multiple polylogarithm function of level two. We find that these integrals can be expressed in terms of multiple zeta (star) values, their related variants (multiple t-values, multiple T-values, multiple S-values, etc.), and multiple harmonic (star) sums and their related variants (multiple T-harmonic sums, multiple S-harmonic sums, etc.), which are closely related to some special types of Schur multiple zeta values and their generalizations. Using these integral identities, we prove many explicit evaluations of Kaneko-Yamamoto multiple zeta values and their related variants. Further, we derive some relations involving multiple zeta (star) values and their related variants.

For k < ℓ, let E_k(z) and E_ℓ(z) be Eisenstein series of weights k and ℓ, respectively, for SL₂(ℤ). We prove that between any two zeros of E_k(e^iθ) there is a zero of E_ℓ(e^iθ) on the interval π/2 < θ < 2π/3.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily non-negative integers. Chinen discovered several families of formal weight enumerators and investigated the validity of the Riemann hypothesis analogue for them. In this paper, the zeta polynomial is computed for Chinen's formal weight enumerators, and a simple criterion is given for the validity of the Riemann hypothesis analogue.

Komori, Matsumoto and Tsumura introduced a zeta function ζ_r (s, Δ) associated with a root system Δ. In this paper, we introduce a q-analogue of this zeta function, denoted by ζ_r (s, a, Δ; q), and investigate its properties. We show that a ‘Weyl group symmetric' linear combination of ζ_r (s, a, Δ; q) can be written as a multiple integral over a torus involving functions Ψ_s. For positive integers k, functions Ψ_k can be regarded as q-analogues of the periodic Bernoulli polynomials. When Δ is of type A₂ or A₃, the linear combinations can be expressed as the functions Ψ_k, which are q-analogues of explicit expressions of Witten's volume formula. We also introduce a two-parameter deformation of the zeta function ζ_r (s, Δ) and study its properties.

The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved that there exist non-standard components if the length is sufficiently large. We give two other proofs of their theorem: one is based on some properties of length functions, and the other is based on the theory of complex projective structures and complex earthquakes. From the latter proof, we can characterize the existence of non-standard components in terms of exotic projective structures with quasi-Fuchsian holonomy.

Kaneko and Tsumura introduced a new kind of multiple zeta function η(k₁, . . . , k_r ; s₁, . . . , s_r). This is an analytic function of complex variables s₁, . . . , s_r, while k₁, . . . , k_r are nonpositive integer parameters. In this paper, we first extend this function to an analytic function η(s'₁, . . . , s'_r ; s₁, . . . , s_r) of 2r complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these η-values and the multiple zeta values ζ(k₁, . . . , k_r) of Euler-Zagier type.