| Budget Amount *help |
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
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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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