论文标题
关于riemann zeta函数的pidduck多项式和零
On Pidduck polynomials and zeros of the Riemann zeta function
论文作者
论文摘要
对于$ 1 <p <\ infty $，我们证明，$ s $的必要条件是riemann zeta Zeta函数的零，$ 0 <\ re s <1 $是$$ \ left（\ begin {array} {array} {cccccc} {ccccc} {ccccc} {ccccc} 1＆\ frac {1＆\ frac {1} {1}＆\ frac {3}＆\ frac {3}＆\ frac {3}＆\ frac {＆} \ frac {1} {7}＆\ frac {1} {9}＆\ cdots \\ - \\ - \ frac {s} {3} {3}＆1＆\ frac {1} {3} {3}＆\ frac {1} - \ frac {s} {5}＆ - \ frac {s} {5}＆1＆\ frac {1} {3} {3}＆\ frac {1} {5} {5}＆\ cdots \\\\\\\\\\\ - \ \ \ \ \ \ freac {s}}} & 1 & \frac{1}{3} & \cdots \\ -\frac{s}{9} & -\frac{s}{9} & -\frac{s}{9} & -\frac{s}{9} & 1 & \cdots \\ \vdots &\vdots & \vdots &\vdots & \vdots & \ddots\\ \end{array}\right)\left(\begin{array}{c} v_0 \\ v_1 \\ v_2 \\ v_3 \\ v_4 \\ \vdots \\ \vdots \\ \end{array}\right) =0 $$ has a非平地解决方案$ \ left（v_ {k} \ right）_ {k = 0}^\ infty $ in $ \ ell^p $。 K. M. Ball在2017年发现了类似的矩阵方程，但目前的论文提供了不同的（独立）的观点。在本文中，$ v_ {k} $的明确公式是根据pidduck多项式构建的。在此过程中，还表明，pidduck多项式形成了与多项式的内部产物$ f，g $相对于正式的基础，我们在正式的表达中替换为“ $ \ sum_ {n = 1}^\ infty（-1） “ $ \ sum_ {n = 1}^\ infty（-1）^{n+1} n^{1+2k} $“ with zeta-function unction junroctional valuep。我们还讨论了可能的非简单零零的修改，并以对零的简单性和与希尔伯特 - 帕利亚计划的关系的应用进行了总结。
For $1<p<\infty$, we prove that a necessary and sufficient condition for $s$ to be a zero of the Riemann zeta function in the strip $0<\Re s<1$ is that $$\left(\begin{array}{cccccc} 1 & \frac{1}{3} & \frac{1}{5} & \frac{1}{7} & \frac{1}{9} & \cdots \\ -\frac{s}{3} & 1 & \frac{1}{3} & \frac{1}{5} & \frac{1}{7} & \cdots \\ -\frac{s}{5} & -\frac{s}{5} & 1 & \frac{1}{3} & \frac{1}{5} & \cdots \\ -\frac{s}{7} &-\frac{s}{7} & -\frac{s}{7} & 1 & \frac{1}{3} & \cdots \\ -\frac{s}{9} & -\frac{s}{9} & -\frac{s}{9} & -\frac{s}{9} & 1 & \cdots \\ \vdots &\vdots & \vdots &\vdots & \vdots & \ddots\\ \end{array}\right)\left(\begin{array}{c} v_0 \\ v_1 \\ v_2 \\ v_3 \\ v_4 \\ \vdots \\ \vdots \\ \end{array}\right) =0 $$ has a nontrivial solution $\left(v_{k}\right)_{k=0}^\infty$ in $\ell^p$. A similar matrix equation was discovered by K. M. Ball in 2017, but the current paper offers a different (and independent) perspective. In this paper an explicit formula for $v_{k}$ is constructed in terms of Pidduck polynomials. In the process, it is also shown that Pidduck polynomials form an orthogonal basis with respect to an inner product of polynomials $f,g$ whereby we replace in a formal expression "$\sum_{n=1}^\infty (-1)^{n+1}n \overline{f(n^2)} g(n^2)$" the divergent sums "$\sum_{n=1}^\infty (-1)^{n+1}n^{1+2k}$" with their zeta-function regularized values. We also discuss the modification for possible non-simple zeros and conclude with applications to the question of the simplicity of the zeros and a relation to the Hilbert-Pólya program.
