Number representations and arithmetical properties of the numbers

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K17504
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Hirosaki University
• Principal Investigator: Tachiya Yohei 弘前大学, 理工学研究科, 准教授 (90439539)
• Research Collaborator: Luca Florian University of the Witwatersrand, Professor ; Coons Michael University of Newcastle, Senior Lecturer ; Elsner Carsten University of Applied Science FHDW, Professor
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 整数論 / 無理数 / 超越数 / ランベルト級数 / テータ関数 / 保型形式

Outline of Final Research Achievements

We studied and obtained a refinement of Erdos's theorem(1948), which gives some structural properties for the q-ary expansion of Lambert series. In particular, we obtained linear independence results for the reciprocal sums of binary recurrences associated with Dirichlet characters. Furthermore we investigated arithmetical properties for some analytic functions and showed that taking an asymptotic viewpoint allows one to prove much stronger transcendence statements in many general situations.
We also studied algebraic properties of the classical theta-constants and gave explicit algebraic dependence relations in some particular cases. This yields algebraic independence results for certain values of the theta-constants and the transcendence results for certain Lambert series.

Free Research Field

数物系科学