Recently I've heard about the Riemann hypothesis for one-variable exponential sums, which states as
For a polynomial $f\in\mathbb{F}_{p^k}[x]$ of degree $d$ and a character $\chi$ of $(\mathbb{F}{p^k},+)$$(\mathbb{F}_{p^k},+)$, provided $(d,p)=1$, we have$$\left|\sum_{x\in\mathbb{F}_{p^k}}\chi(f(x))\right|\le(d-1)\sqrt{p^k}$$
I also know the $L$-function associated to $f$ is$$L(f,T)=\exp\left(\sum_{n\ge1}S_n(f,\chi)\frac{T^n}{n}\right)$$ where $S_n(f,\chi)=\sum_{x\in\mathbb{F}_{p^{kn}}}\chi(\operatorname{tr}_{\mathbb{F}_{p^{kn}}/\mathbb{F}_{p^{k}}}(f(x)))$.
My question is, what is the relation of the Riemann hypothesis for one-variable exponential sums, and Riemann hypothesis for the associated $L$-function? I guess they are equivalent forms, but I can't prove it.
My thought: This is similar to the relation in the elliptic curve version. The Hasse theorem
For an elliptic curve $E$ over $\mathbb{F}_p$, $|\#E(\mathbb{F}_p)-p-1|\le 2\sqrt p$.
is equivalent to the Riemann hypothesis for elliptic curve:
The zeros of $\zeta(E,s)$ has real part $\frac1{2}$.
This equivalence is due to computing the zeta function $\zeta(E,s)$, involving Riemann-Roch theorem. But as for the exponential sum case, I have no idea how to compute the $L$-function associated to $f\in\mathbb{F}_{p^k}[x]$.
Any help will be appreciated.
