2005 Fiscal Year Final Research Report Summary
Analytic study on distribution properties of the residual order and residual index, and on the relation between the distribution and estimates on character sums
| Project/Area Number |
14540048
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Meiji Gakuin University |
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Principal Investigator |
MURATA Leo Meijigakuin University, Dept.of Economics, Professor, 経済学部, 教授 (30157789)
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| Co-Investigator(Kenkyū-buntansha) |
KITAOKA Yoshiyuki Meijyou University, Dept.of Mathematics, Professor, 理工学部, 教授 (40022686)
OKAZAKI Ryutarou Doshisha University, Dept.of Technology, Lecturer, 工学部, 専任講師 (20268113)
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| Project Period (FY) |
2002 – 2005
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| Keywords | residual order / resudual index / Artin's Conjecture on primitive roots / estimate of character sums |
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| Research Abstract |
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".
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