We study some finite discrete groups such as semi-direct products of finite cyclic groups by their automorphisms, the corresponding group and subgroup C* -algebras, and their K-theory. Consequently, we obtain several isomorphism classification theorems of such groups by their group C* -algebras and K-theory.

We introduce an iterated integral version of (generalized) log-sine integrals

(iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals.

For each positive integer m, an arbitrary finite non-solvable group acts smoothly on infinitely many standard spheres with exactly m fixed points. However, for a given finite non-solvable group G and a given positive integer m, all standard spheres do not admit smooth actions of G with exactly m fixed points. In this paper, for each of the alternating group A₆ on six letters, the symmetric group S₆ on six letters, the projective general linear group PGL(2, 9) of order 720, the Mathieu group M₁₀ of order 720, the automorphism group Aut(A₆) of A₆ and the special linear group SL(2, 9) of order 720, we will give the dimensions of homology spheres whose fixed point sets of smooth actions of the group do not consist of odd numbers of points.

We give a formula for the quantum SU(2) invariant at ζ = e^4πi/r for Lens space L(p, q), and we prove that the asymptotic expansion is represented by a sum of contributions from SL₂C flat connections whose coefficients are square roots of the Reidemeister torsions.

Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.

As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185-195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann's fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628-634). We may thus refer to this as the ‘Fourier series-boundary q-series', and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275-289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori, the modular relation

(S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).

We prove an identity for multiple zeta star values, which generalizes some identities due to Imatomi, Tanaka, Tasaka and Wakabayashi. This identity gives an analogue of cyclic insertion-type identities, for multiple zeta star values, and connects the block decomposition with Zhao's generalized 2-1 formula.

In this paper, we study the mean square of the logarithmic derivative of the Selberg zeta function for cocompact discrete subgroups. Our results are analogues of the results on the mean square of the logarithmic derivative of the Riemann zeta function by Goldston, Gonek,and Montgomery (J. Reine Angew. Math. 537 (2001), 105-126). We obtain an asymptotic formula for the mean square of the logarithmic derivative of the Selberg zeta function,including a term on the pair correlation of the zeros of the Selberg zeta function. In addition,we introduce an integral related to the prime geodesic theorem in short intervals and prove that the integral is bounded by the mean square of the logarithmic derivative of the Selberg zeta function. The upper bound for the integral is improved in the case of the Selberg zeta function for arithmetic cocompact groups by proving an asymptotic formula for the mean square near the left side of the vertical line whose real part is one.

We consider the inhomogeneous Dirichlet-boundary value problem for the quadratic nonlinear Schrödinger equations, which is considered as a critical case for the large-time asymptotics of solutions. We present sufficient conditions on the initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques of the free Schrödinger group.

Kaneko and Koike introduced the notion of extremal quasi-modular forms and proposed conjectures on their arithmetic properties. The aim of this paper is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The paper ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalized extremal quasi-modular form of weight 14 and depth one.

The Fisher integral is the normalizing constant of a statistical model on the special orthogonal group. In this paper, we discuss a system of differential equations for the Fisher integral. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we show that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. The Fisher integral for diagonal matrices is related to the hypergeometric function of matrix arguments. We also give a new approach to get differential operators annihilating the Fisher integral for the diagonal matrix.

Let k₁, . . . , k_r be positive integers. Let q₁, . . . ,q_r be pairwise coprime positive integers with q_i > 2 (i = 1, . . . , r), and set q = q₁ . . . q_r. For each i = 1, . . . , r, let T_i be a set of φ(q_i)/2 representatives mod q_i such that the union T_i ∪ (-T_i) is a complete set of coprime residues mod q_i. Let K be an algebraic number field over which the qth cyclotomic polynomial Φ_q is irreducible. Then, φ(q)/2 ^r numbers

Π^r_i=1d ^ki-1/dz ^ki-1_i (cotπz_i)|_zi=ai/qi (a_i ∈ T_i , i = 1, . . . , r)

are linearly independent over K. As an application, a generalization of the Baker-Birch-Wirsing theorem on the non-vanishing of the multiple Dirichlet series L(s₁, . . . , s_r; f) with periodic coefficients at (s₁, . . . , s_r) = (k₁, . . . , k_r) is proven under a parity condition.

We study the rationality of gamma factors associated to certain Hasse zeta functions. We show many explicit examples of rational gamma factors coming from products of GL(n).