Extension of almost periodic functions and the distribution of zeros

利用遍历理论和函数代数理论，把狄利克雷级数看作玻尔群上的解析函数，研究了狄利克雷级数的性质。结合解析数论，我们特别关注黎曼ζ-函数的情况，得到了一个弱意义上的中值定理和关于ζ-函数值分布的一些结果。设K为离散群{log γ；γ正有理}。则自然定义了K的同胚的一个单参数群{T_t}_<t∈R>。固定1/2 < u和putZ_u(x) = Σ^^∞___<n=1>(1)/(n^u)x_<log n>^(x), x∈K，其中x_<log n>表示以log n表示的字符。则t→Z_u（T_tO）表示ζ(u+it), Z_u(x)是H^2(K)的外函数。由于（K, {T_t}_<t∈R>）是一个遍历流，我们有如下黎曼ζ函数的中值定理：定理设0 < K <∞，且设l > 0。然后是一个子集的Z ^ + J密度为零,< lim > ___ < J【∋! /】N→∞> (1)/ (Nl)Σ^ ^ ^ < N - 1 > ___ < N = 0 >∫^ ^ ^ < (N + 1) l > ___ <问> |ζ(u +) | ^ < 2 k > dt =∫_K | Z_u (x) | ^ < 2 k > dσ(x)。这个定理表明林德洛夫假设在弱意义上成立。我们还研究了所有Dirichlet级数t→Z_u（T_t x）与欧拉积的类。这使我们能够理解黎曼ζ函数的特性。

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 putZ_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.