| Co-Investigator(Kenkyū-buntansha) |
NAKAO Shintaro Faculty of Science Full Professor, 理学部, 教授 (90030783)
ICHINOSE Takashi Faculty of Science Full Professor, 理学部, 教授 (20024044)
FUJIMOTO Hirotaka Faculty of Science Full Professor, 理学部, 教授 (60023595)
TAMURA Hiroshi Faculty of Science Associate Professor, 理学部, 助教授 (80188440)
FUJIMAGARI Tetsuo Faculty of Science Full Professor, 理学部, 教授 (60016102)
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| Research Abstract |
Our purpose of the project is as follows : We formulate Dirichlet's density theorem stating the probability of two integers to be co-prime as law of large numbers (LLN), and then we consider central limit theorem scaling (CLT-scaling) and find a limit theorem on it.
Let us consider (Zhat, ＼lambda) as a fundamental probability space, where Zhat is a finite integral adele and ＼lambda the Haar probability measure on it. For each (x,y) ＼in Zhat ＼times Zhat, let X(x,y) = 1 or 0 according as (x,y) is co-prime or not. Then, as N＼to＼infty, S_N(x,y) = (1/N)^2 ＼sum_{m,n=1}^N X(x+m,y+n) converges to 6/＼pi^2 a.s., which is just LLN.
Next we consider the limit behavior of CLT-scaling N(S_N(x,y)-6/＼pi^2). Then we can describe completely the set of all limit points of {N(S_N(x,y)-6/＼pi^2)} in the L^2-space by parametrizing them continuously in terms of elements of a quotient ring Zhat/＼sim. In particular, N(S_N(x,y)-6/＼pi^2) is not convergent as N＼to＼infty. In a word, CLT does not hold!
If, however, we
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interpret the convergence in the sense of Cesaro, then
(1/N) ＼sum_{n=l}^N n(S_n(x,y}-6/＼pi^2) ＼to U(x) + U(y) in l^2.
Here
U(x) = ＼sum_{u=1}^{＼infty} (＼mu(u)/u) ((x＼mod u)/u - (u-1)/2u) in L^2,
where ＼mu(u) is the Mobius function. So our study turns to an investigation of this U. In this project, it is seen that the distribution of U is symmetric and has moments of all orders. We further expect that U will be not normal distributed, although normal distributions with mean zero have the property above. If this is proved, we want to call the convergence above non CLT.
On the one hand, from the description of limit points of {N(S_N(x,y) -6/＼pi^2)} N_k(S_{N_K}(x,y)-6/＼pi^2) ＼to 0 in L^2 for whatever subsequence {N_k} such that N_k ＼not=0 and N_k ＼to 0 in Zhat/＼sim. Renormalizing this by its standard deviation in order to find a nontrivial limit, we expect that the renormalization will converge to a standard normal distribution.
We can not succeed in proving these two conjectures within the term of project. We are instead giving a verification by computational experiment. Less
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