Jacobi symbol

DEFINITION 7.1   Let $m$ be a positive odd integer. Let us factor $m$:

$$m=\prod_i p_i^{e_i}$$

where $p_i$ are primes. Then for any $n\in \mathbb{Z}$, we define Jacobi symbols as follows

$${\left(\frac{n}{m}\right)}= \prod_i{\left(\frac{n}{p_i}\right)}^{e_i}$$

We further define

$${\left(\frac{a}{p}\right)}= 0$$$$\text { if } a \in p\mathbb{Z}.$$

THEOREM 7.2 (quadratic reciprocity theorem)   For any positive odd integers $n,m$, we have

$$\left(\frac{m}{n}\right) \left(\frac{n}{m}\right) =(-1)^{(m-1)(n-1)/4}.$$

THEOREM 7.3   Let $n$ be a postive odd integer. Then:

1. ${\left(\frac{-1}{m}\right)}=(-1)^{(m-1)/2}$.
2. ${\left(\frac{2}{m}\right)}=(-1)^{(m^2-1)/8}$.

EXERCISE 7.1   $p=113357$ is a prime. (You may use the fact without proving it.) Is there any integer $n$ such that

$$n^2=11351$$    in $$\mathbb{Z}/p \mathbb{Z}$$$$\text { ?}$$

If so, can you find such $n$?

A litte appendix. (The following is borrowed from Wikipedia(“Riemann zeta function”,Japanese version,May 2019))

$$\zeta(s)=\sum_{s=1}^\infty \frac{1}{n^s}=\prod (1-\frac{1}{p^s})^{-1}$$

$$\log(\zeta(s)) = -\sum_p \log(1-\frac{1}{p^s} )$$

$$= \sum_{p} \sum_{n=1}^\infty \frac{1}{n p^{ns}}$$

$$= \sum_{p} \sum_{n=1}^\infty \frac{1}{n p^{ns}}$$

$$= s \sum_{n=1}^\infty \frac{1}{n} \sum_p \int_{p^n}^\infty x^{-s-1} dx$$

$$= s \sum_{n=1}^\infty \frac{1}{n} \int_{1}^\infty (\sum_p [? p\leq x^{1/n}]) x^{-s-1} dx$$

$$= s \sum_{n=1}^\infty \frac{1}{n} \int_{1}^\infty \pi(x^{1/n}) x^{-s-1} dx$$

$$= \int_1^\infty \Pi(x) x^{-s-1} dx$$

here we have put

$$\pi(x)=\char93 \{p; p\leq x\} , \qquad \Pi(x)=\sum_{n=1}^\infty \frac{1}{n} \pi (x^{1/n}).$$

Note also that we have used $\frac{1}{p^{ns}}=s\int_{p^n}^\infty {x^{-s-1}} dx .$