Second main terms for density theorems in number theory

2015 Fiscal Year Final Research Report

• Project/Area Number: 24654005
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kobe University
• Principal Investigator: Taniguchi Takashi 神戸大学, 理学(系)研究科(研究院), 准教授 (60422391)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 第二主要項 / 密度定理 / 三次体 / 概均質ベクトル空間 / 指数和

Outline of Final Research Achievements

We studied in details the second main terms in counting discriminants of certain families of number fields. The results include: On the counting function for non-commutative sextic Galois fields, we get an evidence that there might exist the tertiary main term. On the counting function for relative cubic extensions, when the base field is a quadratic field, we determined the second main term; over a general number field, we determined the second main term for a smoothed counting function. We have several more results. As a related problem, we discover a simple method to evaluate certain exponential sums associated with prehomogeneous vector spaces.

Free Research Field

整数論