Riemann Xi-函数的实根Pólya-样逼近
Real-rooted Pólya-like approximations to the Riemann Xi-function
摘要：
黎曼$Ξ(z)$函数允许偶核$Φ(t)$的傅里叶变换。后者与Jacobi theta函数$θ(z)$的导数有关，$1/2$是权重的模形式。Pólya注意到，当$t$变为无穷大时，$e^t$变为$e^t+ e^{-t}=2\cosh t$。然后，他近似的核心$Φ(t)$的$Φ_{P}(t)$只包含领先的长期和$\exp t,\exp(9t/4)$取代$2\cosh t,2\cos(9t/4)$。该过程捕获了来自尾部的几乎所有贡献（即，$t\to\infty$）。我们意识到当$t$变为无穷大时，$0\leqslant b<1,c\in\R$，$\cosh t+c \cosh(bt)$变为$\cosh t$。因此，我们通过用$\cosh(9t/4)+b\sum_{k=0}^{m-1}b_k \cosh(9kt/(4m))$代替$\cosh(9t/4)$并调整参数$b,b_k,m$来改进Pólya的近似，使得（A）当$t$变为无穷大时，近似核$Φ_{S}(b,b_k,m;t)$变为$Φ(t)$;（B）在$t=0$处，$Φ_{S}(b,b_k,m;t)$与$Φ(t)$相同;（C）与Pólya的情况一样，$Φ_{S}(b,b_k,m;t)$的傅里叶变换只有实零点。由于该过程还捕获了来自头部的几乎所有贡献（即，接近$t=0$），我们能够锚定内核$Φ(t)$的两端。
英文摘要：
The Riemann $Ξ(z)$ function admits a Fourier transform of a even kernel $Φ(t)$. The latter is related to the derivatives of Jacobi theta function $θ(z)$, a modular form of weight $1/2$. Pólya noticed that when $t$ goes to infinity, $e^t$ goes to $e^t+ e^{-t}=2\cosh t$. He then approximated the kernel $Φ(t)$ by $Φ_{P}(t)$ that contained only the leading term and with $\exp t,\exp(9t/4)$ replaced by $2\cosh t,2\cos(9t/4)$. This procedure captured almost all of the contribution from the tail part (i.e., $t\to\infty$) of the kernel $Φ(t)$. We realize that when $t$ goes to infinity and $0\leqslant b<1,c\in\R$, $\cosh t+c \cosh(bt)$ goes to $\cosh t$. Thus we improve Pólya's approximation by replacing $\cosh(9t/4)$ with $\cosh(9t/4)+b\sum_{k=0}^{m-1}b_k \cosh(9kt/(4m))$ and adjusting the parameters $b,b_k,m$ such that (A) the approximated kernel $Φ_{S}(b,b_k,m;t)$ goes to $Φ(t)$when $t$ goes to infinity;(B) $Φ_{S}(b,b_k,m;t)$ is identical to $Φ(t)$ at $t=0$; (C) the Fourier transform of $Φ_{S}(b,b_k,m;t)$,like in Pólya's case, has only real zeros. Since this procedure also captures almost all of the contribution from the head part (i.e., near $t=0$) of the kernel $Φ(t)$, we are able to anchor both ends of the kernel $Φ(t)$.
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