Shigeki Akiyama
University of Tsukuba
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Publication
Featured researches published by Shigeki Akiyama.
Ramanujan Journal | 2001
Shigeki Akiyama; Yoshio Tanigawa
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.
Theoretical Computer Science | 2002
Shigeki Akiyama; Attila Peth ohuml; ohuml
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.
Archive | 1998
Shigeki Akiyama
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.
Mathematics of Computation | 1999
Shigeki Akiyama; Yoshio Tanigawa
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.
Archive | 2002
Shigeki Akiyama; Hideaki Ishikawa
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.
Archive | 2015
Shigeki Akiyama; M. Barge; Valérie Berthé; J.-Y. Lee; A. Siegel
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.
Mathematical Proceedings of the Cambridge Philosophical Society | 2004
Jun Luo; Shigeki Akiyama; Jörg M. Thuswaldner
In this paper we consider topological properties of ordinary as well as graph directed iterated function systems. First we give the basic definitions.
Osaka Journal of Mathematics | 2008
Shigeki Akiyama; Horst Brunotte; Attila Pethö; Jörg M. Thuswaldner
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
Archive | 2006
Shigeki Akiyama
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.
Journal of Mathematical Analysis and Applications | 2003
Shigeki Akiyama; Horst Brunotte; Attila Pethő
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.