These are warm up problems that do not need to be turned in.
(a) In class we gave an elementary proof that $\vartheta(x)=O(x)$. Give a similarly elementary proof that $x=O(\vartheta(x))$ (both bounds were proved by Chebyshev before the PNT).
(b) Prove the Möbius inversion formula, which states that if $f$ and $g$ are functions $\mathbb{Z}_{\geq 1} \rightarrow \mathbb{C}$ that satisfy $g(n)=\sum_{d \mid n} f(d)$ then $f(n)=\sum_{d \mid n} \mu(d) g(n / d)$, where $\mu(n):=(-1)^{\#\{p \mid n\}}$ if $n$ is squarefree and $\mu(n)=0$ otherwise.
(c) Verify that for all Schwartz functions $f, g \in \mathcal{S}(\mathbb{R})$ we have
$$\widehat{f * g}=\hat{f} \hat{g}, \quad \text { and } \quad \widehat{f g}=\hat{f} * \hat{g} .$$
(the Fourier transform turns convolutions into products and vice versa).

\begin{proof}

(a) 通过比较素数计数函数ϑ(x)和x，利用比例原理得出x = O(ϑ(x))。 (b) 用数论函数和莫比乌斯反演公式的定义，然后证明等式。 (c) 用傅立叶变换的性质，分别证明傅立叶变换的乘法定理和卷积定理。

\begin{proof}

E-mail: help-assignment@gmail.com  微信:shuxuejun

help-assignment™是一个服务全球中国留学生的专业代写公司