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Zeroes and zero-free regions

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Bounds on the Dedekind zeta-function

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The knowledge on the general Dedekind zeta is less accomplished than the one of the Riemann zeta-function, but we still have interesting results. Theorem 4 of gives the convexity bound. See also section 4.1 of .
Theorem (1959)

In the strip $-\eta\le \sigma\le 1+\eta$, $0 < \eta\le 1/2$, the Dedekind zeta function $\zeta_K(s)$ belonging to the algebraic number field $K$ of degree $n$ and discriminant $d$ satisfies the inequality $$ |\zeta_K(s)|\le 3 \left|\frac{1+s}{1-s}\right| \left(\frac{|d||1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}} \zeta(1+\eta)^n. $$

We denote by $N_K(T)$ the number of zeros $\rho$, of the Dedekind zeta-function of the number field $K$ of degree $n$ and discriminant $d_K$, zeros that lie in the critical strip $0 < \Re \rho = \sigma < 1$ and which verify $|\Im \rho|\le T$. After a first result in , we find in the following result.
Theorem (2014)

When $T\ge1$, we have $N_K(T)=\frac{T}{\pi}\log\Bigl(|d_K|\Big(\frac{T}{2\pi e}\Bigr)^n\Bigr) +O^*\bigl(0.316(\log |d_K|+n\log T)+5.872 n+3.655\bigr)$ .

This is improved in into:
Theorem (2021)

When $T\ge1$, we have $N_K(T)=\frac{T}{\pi}\log\Bigl(|d_K|\Big(\frac{T}{2\pi e}\Bigr)^n\Bigr) +O^*\bigl(0.228(\log |d_K|+n\log T)+23.108 n+4.520\bigr)$ .

In , a zero-free region is proved.
Theorem (1959)

Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and of discriminant $d \ge 2$. The associated Dedekind zeta-function $\zeta_K$ has no zeros in the region $$ \sigma\ge 1-\frac{1}{12.55\log|d_K|+n(9.69\log|t|+3.03)+58.63}, |t|\ge1 $$ and at most one zero in the region $$ \sigma\ge 1-\frac{1}{12.74\log|d_K|}, |t|\le 1. $$ The exceptional zero, if it exists, is simple and real.

See for a result for Hecke $L$-series.

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