Analytic study on distribution properties of the residual order and residual index, and on the relation between the distribution and estimates on character sums

残差阶数和残差指数的分布特性以及分布与特征和估计之间关系的解析研究

• 批准号: 14540048
• 负责人: MURATA Leo
• 金额: $ 2.05万
• 依托单位: Meiji Gakuin University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540048/
• 关键词: residual order; resudual index; Artin's Conjecture on primitive roots; estimate of character sums; 剰余指数; 剰余位数; 原始根に関するArtinの予想

Let a be a natural number, p be a prime number, D_a(p) and I_a(p) denote the residual order and the residual index of the class a(mod p), respectively.Both D_a(p) and I_a(p) are surjective map from P(the set of all primes) to N(the set of all natural numbers). We studied here the distribution properties of the map D_a(p).Firstly, we considered the set : Q_a(x ; s, t){p≦x ; D_a(p)≡s (mod t)}, s and t be natural numbers, and we obtained the results ;1)when we assume the generalized Riemann Hypothesis, we can prove the existence of the natural density Δ_a(s,t) of the set Q_a(x ; s, t),2)we can calculate the density Δ_a(s, t) effectively, and through some numerical and theoretical observations, we found out some interesting properties of the number theoretical function Δ_a(s, t).Secondly, we considered the set : M(x) = {p ≦ x ; D_2(p) is a prime}. Then, under G.R.H., we have an estimate #M(x) << x(log x)^(-2). This estimate is most likely to be "best possible".

设a是自然数，p是素数，D_a(P)和I_a(P)分别表示A(Modp)类的剩余阶和剩余指数，D_a(P)和I_a(P)都是从P(所有素数的集合)到N(所有自然数的集合)的满射.本文研究了映射D_a(P)的分布性质.首先，考虑集合Q_a(x；S，t){p≦x；D_a(P)≡S(Mod T)}，S和t是自然数，得到了如下结果：1)当我们假设广义黎曼假设时，我们可以证明集合Q_a(x，t)的自然密度Δ_a(S，t)的存在性；S，t)，2)我们可以有效地计算Δ_a(S，t)的密度，通过一些数值和理论观察，我们发现了数论函数Δ_a(S，t)的一些有趣的性质。其次，我们考虑了集合：m(X)={p≦x；D_2(P)是素数}。然后，在G.R.H.下，我们有一个估计#M(X)&lt；&lt；x(Logx)^(-2)。这一估计很可能是“最好的可能”。

项目成果

On the largest prime factor of a Mersenne number.

关于梅森数的最大素因数。

• 发表时间: 2004
• 期刊: Center de Recherches Mathematiques, CRM Prceedings and Lecture Notes Volume 36
• 作者: Leo Murata;K. Chinen;Leo Murata (with Koji Chinen);Leo Murata (with Koji Chinen);Leo Murata (with Carl Pomerance)
• 通讯作者: Leo Murata (with Carl Pomerance)

On the largest prime factor of a Mersenne number

关于梅森数的最大素因数

• 发表时间: 2004
• 期刊: Centre de Recherches Mathematiques, CRM Proceedings and Lecture Notes 36
• 作者: Leo Murata;C. Pomerance
• 通讯作者: C. Pomerance

On a distribution property of a residual order of a(mod p).

关于a(mod p)的残差阶数的分布性质。

• 发表时间: 2003
• 期刊: Proc.of Japan Acad. Vol.79
• 作者: Leo Murata;K. Chinen;Leo Murata (with Koji Chinen)
• 通讯作者: Leo Murata (with Koji Chinen)

ON A DISTRIBUTION PROPERTY OF THE RESUDUAL ORDER OF a (mod p)-IV

论a(mod p)-IV的余阶的分布性质

• 期刊: Number Theory. Tradition and Modernization(Edited by Y.Tanigawa and W.Zhang)(Springer Veerlag) to appear
• 作者: Leo Murata;K. Chinen;Leo Murata (with K.Chinen);Leo Murata (with K.Chinen)
• 通讯作者: Leo Murata (with K.Chinen)

Leo Murata, K.Chinen: "On a distribution property of the residual order of a (mod p)"Journal of Number Theory. 105. 60-81 (2004)

Leo Murata，K.Chinen：“关于 a (mod p) 的残差阶的分布性质”数论杂志。