论文标题
希尔伯特太空基础和希尔伯特的第八个问题
The Hilbert space basis and Hilbert's eighth problem
论文作者
论文摘要
本文认为，在任何间隔$ \ {（a，a，b）_r \} _ {r = 1}^{r = 1}^{\ in \ in \ in \ in \ mathbb {r rathbb {r rathbb {r rathbb {r} $上。根据$ \ hat {h} _r $中的真实正交多项式的零的定理所示，存在一个完整的正交基础$ \ {f（x）_k \} _ {k = 1} $ f（x）\ in \ {f（x）_k \} _ {k = 1}^{\ infty} $具有零，那么这些零是简单而真实的。总体化的Hardy函数$ z（σ，t）= \reζ（σ+it）e^{iθ（t）} $。显示出在希尔伯特太空中$ \ hat {h} _r $存在完整的基础$ \ {z（λ_k，t \} _ {k = 1}^{\ infty} $ whend $ z（t）\ in \ {z（λ_k，t \} _ {k = 1}^{\ infty} $时，$λ_k= 1/2 $，因此hardy函数$ z（t）=ζ（1/2+it）
The paper considers the Hilbert space $\hat{H}_r$ of real functions summable with the square $L^2(a,b)_r$ on any interval $\{(a,b)_r\}_{r=1}^{\infty}\in \mathbb{R}$. It is shown on the basis of the theorem on zeros of real orthogonal polynomials if in $\hat{H}_r$ there exists a complete orthonormal basis $\{f(x)_k\}_{k=1}^{\infty}$ and the function $f(x)\in\{f(x)_k\}_{k=1}^{\infty}$ has zeros, then these zeros are simple and real. The generalized Hardy function $Z(σ,t)=\Reζ(σ+it)e^{iθ(t)}$ is considered. It is shown that in the Hilbert space $\hat{H}_r$ there exists a complete basis $\{Z(λ_k,t\}_{k=1}^{\infty}$ where $λ_k\in\mathbb{Q}$ and $Z(t)\in\{Z(λ_k,t\}_{k=1}^{\infty}$ when $λ_k=1/2$, hence the Hardy function $Z(t)=ζ(1/2+it)e^{iθ(t)}$ has all simple and real zeros.
