# 统计代写|随机过程代写stochastic process代考|MATH3801

#### Doug I. Jones

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## 统计代写|随机过程代写stochastic process代考|Perturbed Version of the Riemann Hypothesis

When I first investigated Poisson-binomial processes, it was to study the behavior of some mathematical functions represented by a series. The idea was to add little random perturbations to the index $k$ in the summation, in short, replacing $k$ by $X_k$, turning the mathematical series into a random function, and see what happens. Here this idea is applied to the Riemann zeta function [Wiki]. The purpose is to empirically check whether the Riemann Hypothesis [Wiki] still holds under small perturbations, as non-trivial zeros of the Riemann zeta function $\zeta$ are very sensitive to little perturbations. Instead of working with $\zeta(z)$, I worked with its sister, the Dirichlet eta function $\eta(z)$ with $z=\sigma+i t \in \mathbb{C}$ [Wiki]: it has the same non-trivial zeros in the critical strip $\frac{1}{2}<\sigma<1$, and its series converges in the critical strip, unlike that of $\zeta$. Its real and imaginary parts are respectively equal to
\begin{aligned} & \Re[\eta(z)]=\Re[\eta(\sigma+i t)]=-\sum_{k=1}^{\infty}(-1)^k \frac{\cos (t \log k)}{k^\sigma} \ & \Im[\eta(z)]=\Re[\eta(\sigma+i t)]=-\sum_{k=1}^{\infty}(-1)^k \frac{\sin (t \log k)}{k^\sigma} \end{aligned}
Note that $i$ represents the imaginary unit, that is $i^2=-1$. I investigated two cases: $\sigma=\frac{1}{2}$ and $\sigma=\frac{3}{4}$. I used a Poisson-binomial process with intensity $\lambda=1$, scaling factor $s=10^{-3}$ and a uniform $F$ to generate the $\left(X_k\right)^{\prime}$ ‘s and replace the index $k$ by $X_k$ in the two sums. I also replaced $(-1)^k$ by $\cos \pi k$. The randomized (perturbed) sums are
\begin{aligned} & \Re\left[\eta_s(z)\right]=\Re\left[\eta_s(\sigma+i t)\right]=-\sum_{k=1}^{\infty} \cos \left(\pi X_k\right) \cdot \frac{\cos \left(t \log X_k\right)}{X_k^\sigma} \ & \Im\left[\eta_s(z)\right]=\Re\left[\eta_s(\sigma+i t)\right]=-\sum_{k=1}^{\infty} \cos \left(\pi X_k\right) \cdot \frac{\cos \left(t \log X_k\right)}{X_k^\sigma} \end{aligned}

Proving the convergence of the above (random) sums is not obvious. The notation $\eta_s$ emphasizes the fact that the $\left(X_k\right)$ ‘s have been created using the scaling factor $s$; if $s=0$, then $X_k=k$ and $\eta_s=\eta$.

Figure 8 shows the orbits of $\eta_s(\sigma+i t)$ in the complex plane, for fixed values of $\sigma$ and $s$. The orbit consists of the points $P(t)=\left(\Re\left[\eta_s(\sigma+i t)\right], \Im\left[\eta_s(\sigma+i t)\right]\right)$ with $0<t<200$, and $t$ increasing by increments of $0.05$. The plots are based on a single realization of the Poisson-binomial process. The sums converge very slowly, though there are ways to dramatically increase the convergence: for instance, Euler’s transform [Wiki] or Borwein’s method [Wiki]. I used $10^4$ terms to approximate the infinite sums.

## 统计代写|随机过程代写stochastic process代考|Dirichlet Eta Function

Let $z=\sigma+$ it be a complex number, with $\sigma$ the real part, and $t$ the imaginary part. The Dirichlet eta function $\eta(z)$ provides an analytic continuation [Wiki] of the Riemann series $\zeta(z)$ in the complex plane $(z \in \mathbb{C})$. The two functions are defined as:
\begin{aligned} & \zeta(z)=\sum_{k=1}^{\infty} \frac{1}{k^s}, \quad \sigma=\Re(z)>1 \ & \eta(z)=\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^s}, \quad \sigma=\Re(z)>0 . \end{aligned}

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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