2001 Fiscal Year Final Research Report Summary
Extension of almost periodic functions and the distribution of zeros
| Project/Area Number |
10640180
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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 |
Basic analysis
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| Research Institution | Waseda University |
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Principal Investigator |
TANAKA Junichi Waseda University, School of Education, Professor, 教育学部, 教授 (60124864)
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| Project Period (FY) |
1998 – 2001
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| Keywords | Dirichlet series / Alomost periodic functions / Riemann ζ-function / Hardy spaces / Distribution of zeros / Mean-value theorems |
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| Research Abstract |
Using the ergodic theory and the theory of function algebras, we investigate the property of Dirichlet series by regarding as analytic functions on Bohr group. In connection with analytic number theory, we especially restrict our attention to the case of the Riemann ζ-function and obtain a mean-value theorem in a weak sense and some results on value distribution of ζ-function.
Let K be the dual group of the discrete group {log γ ; γ positive rational}. Then a one-parameter group {T_t}_<t∈R> of homeomorphisms of K is defined naturally. Fix 1/2 < u and put
Z_u(x) = Σ^^∞___<n=1>(1)/(n^u)x_<log n>^(x), x∈K,
where x_<log n> denotes the character by log n. Then t → Z_u(T_tO) represents ζ(u+it), and Z_u(x) is an outer function of H^2(K). Since (K, {T_t}_<t∈R>) is an ergodic flow, we have the following mean-value theorem of the Riemann ζ-function : Theorem Let 0 < k < ∞, and let l > 0. Then there is a subset J of Z^+ of density zero such that
<lim>___<J【∋!/】N→∞>(1)/(Nl)Σ^^^<N-1>___<n=0>∫^^^<(n+1)l>___<nl>|ζ(u+it)|^<2k>dt=∫_K|Z_u(x)|^<2k>dσ(x). This theorem shows that Lindelof hypothesis holds in a weak sense. We also study the class of all Dirichlet series t → Z_u(T_t x) with Euler products. This enables us to understand the peculiarity of the Riemann ζ-function.
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