Shigeki Akiyama

Multiple zeta values at non-positive integers

Values of Euler-Zagiers multiple zeta function at non-positive integers are studied, especially at (0,0,...,−n) and (−n,0,...,0). Further we prove a symmetric formula among values at non-positive integers.

On canonical number systems

Let P(x)=pdxd++p0Z[x] be such that d1,pd=1,p02 and N={0,1,...,p01}. We are proving in this note a new criterion for the pair {P(x),N} to be a canonical number system. This enables us to prove that if p2,...,pd1,i=1dpi0 and p0>2i=1d|pi|, then {P(x),N} is a canonical number system.

Pisot numbers and greedy algorithm

We study the greedy expansion of real numbers in Pisot number base. We will show a certain criterions of finiteness, periodicity, and purely periodicity. Further, it is proved that every sufficiently small positive rational numbers has purely periodic greedy expansion in Pisot unit base under a certain finiteness condition.

Calculation of values of L -functions associated to elliptic curves

We calculated numerically the values of L-functions of four typical elliptic curves in the critical strip in the range Im(s) 0, which enables us to calculate for large Im(s). Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

On Analytic Continuation of Multiple L-Functions and Related Zeta-Functions

A multiple L-function and a multiple Hurwitz zeta function of EulerZagier type are introduced. Analytic continuation of them as complex functions of several variables is established by an application of the Euler-Maclaurin summation formula. Moreover location of singularities of such zeta functions is studied in detail.

On the Pisot Substitution Conjecture

Our goal is to present a unified and reasonably complete account of the various conjectures, known as Pisot conjectures, that assert that certain dynamical systems arising from substitutions should have pure discrete dynamical spectrum. We describe the various contexts (symbolic, geometrical, arithmetical) in which substitution dynamical systems arise and review the relevant properties of these systems. The Pisot Substitution Conjecture is stated in each context and the relationships between these statements, and with several related conjectures, are discussed. We survey the special cases in which the Pisot Substitution Conjecture has been verified and present algorithmic procedures for checking pure discrete spectrum. We conclude with a discussion of possible extensions to higher dimensions.

On the boundary connectedness of connected tiles

In this paper we consider topological properties of ordinary as well as graph directed iterated function systems. First we give the basic definitions.

Generalized radix representations and dynamical systems III

For r = (r1, . . . , rd) ∈ Rd the map τr : Zd → Zd given by τr(a1, . . . , ad) = (a2, . . . , ad,−br1a1 + · · ·+ rdadc) is called a shift radix system if for each a ∈ Zd there exists an integer k > 0 with τk r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like β-expansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials

Positive Finiteness of Number Systems

We characterize the set of β’s for which each polynomial in β with nonnegative integer coefficients has a finite admissible expression in some number systems.

Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert

A conjecture of W.J. Gilberts on canonical number systems which are defined by cubic polynomials is partially proved, and it is shown that the conjecture is not complete. Applications to power integral bases of simplest and pure cubic number fields are given thereby extending results of S. Kormendi.