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Bounds on the Dedekind zeta-function
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
$$
|\zeta_K(s)|\le 3 \left|\frac{1+s}{1-s}\right|
\left(\frac{|d||1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}}
\zeta(1+\eta)^n.
$$
An explicit approximation of the number of ideals in a number field
was given in the PhD memoir
of J.S. Sunley. It is recalled in Theorem 1.1 of
and further refined there in Theorem 1.2.
Theorem (2022)
For $x > 0$ and $n_K\ge 0$, the number $I_K(x)$ of integral ideals
of norm at most $x$ in the
number field $K$ of degree $n_K$ and discriminant $\Delta_K$ is approximated given by
$$
I_K(x)
=\kappa_K x
+\mathcal{O}^*\biggl(C(K)|\Delta_K|^{\frac{1}{n_K+1}}(\log|\Delta_K|)^{n_K-1}
x^{1-\frac{2}{n_K+1}}\biggr)
$$
where
$$
C(K)=0.17\frac{6n_K-2}{n_K-1}
2.26^{n_K} e^{4n_K+(26/n_K)}n_K^{n_K+(1/2)}
\biggl(
44.39\biggl(\frac{82}{1000}\biggr)^{n_K}n_K!
+
\frac{13}{n_K-1}
\biggr).
$$
For $n_K=2$, the constant $\Delta_K$ is about $8.80\cdot 10^{11}$. The
constant arising from Sunley's work was about $1.75\cdot 10^{30}$.
For $n_K=3$, the constant $C(K)$ is approximately equal to $8.45\cdot 10^{11}$. The
constant arising from Sunley's work was about $8.57\cdot 10^{44}$.
An approximation not relying on the discriminant but on the regulator
and the class number has been given in Corollary 2 of
.
Theorem (2019)
The notation being as above, we have
$$
I_K(x)=\kappa_K x
+O^*\biggl(n_K^{10n_K^2}
(\text{Reg}_Kh_K)^{1/n_K}\biggr)
(1+\log\text{Reg}_Kh_K)^{\frac{(n_K-1)^2}{n_K}}
x^{1-\frac{1}{n_K}}.
$$
The reader will also find there an explicit bound of similar strength
on the number ideals in a given ideal class.
The number of integral ideals in a given ray class is approximated
explicitely following the same scheme in Theorem 1 of
.
3. Bounds for the residue of the Dedekind zeta-function
Let $K$ be a number field over $\mathbb{Q}$ with degree $n_K$ and
discriminant $\Delta_K$. Furthermore, suppose that the residue of the
Dedekind zeta function $\zeta_K(s)$ at $s=1$ is denoted
$\kappa_K$. Unconditional bounds for the residue of the Dedekind
zeta-function at $s=1$ are found in
and in
Section 3 of
.
Theorem (2000, 2022)
If $n_K\geq 3$, then
$$
\frac{0.0014480}{n_K g(n_K){|\Delta_K|}^{1/n_K}}
< \kappa_K \leq
\left(\frac{e\log |\Delta_K| }{2(n_K - 1)}\right)^{n_K - 1},
$$
in which $g(n_K)=1$ if $K$ has a normal tower over $\mathbb{Q}$ and
$g(n_K) = n_K!$ otherwise.
If the Generalised Riemann Hypothesis and Dedekind Conjecture
(i.e. $\zeta_K/\zeta$ is entire) are
true, then stronger bounds are found in Corollary 2 of
.
Theorem (2022)
Assume that the Generalised Riemann Hypothesis and the Dedekind
Conjecture are true. If $n_K\geq 2$, then
$$
\frac{e^{-17.81(n_K -1)}}{\log\log{|\Delta_K|}} \leq \kappa_{K}
\leq e^{17.81(n_K -1 )} (\log\log{|\Delta_K|})^{n_K-1} .
$$
4. Zeroes and zero-free regions
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 (2012)
Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and of
discriminant $d_K$ such that $|d_K| \ge 2$. 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.
This is improved in
into:
Theorem (2021)
Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and of
discriminant $d_K$ such that $|d_K| \ge 2$. The associated Dedekind
zeta-function $\zeta_K$ has no zeros in the region
$$
\sigma\ge 1-\frac{1}{12.2411\log|d_K|+n(9.5347\log|t|+0.0501)+2.2692}, |t|\ge1
$$
and if $d_K$ is sufficiently large, then there is at most one zero in the region
$$
\sigma\ge 1-\frac{1}{12.4343\log|d_K|}, |t|< 1.
$$
The exceptional zero, if it exists, is simple and real.
See
for a result for Hecke $L$-series.
In
In
,
a zero-free region is proved. Here is slightly simplified version of
his result.
Theorem (2017)
Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and of
discriminant $d_K$ such that $|d_K| \ge 8$. The associated Dedekind
zeta-function $\zeta_K$ has no zeros in the regions
$$
\sigma\ge 1-\frac{1}{1.7\log|d_K|}, |t|\ge\frac{1}{4\log|d_K|},
$$
$$
\sigma\ge 1-\frac{1}{2\log|d_K|}, |t|\ge\frac{1}{2\log|d_K|},
$$
and
$$
\sigma\ge 1-\frac{1}{1.62\log|d_K|}, t=0.
$$
In
,
we find the next result.
Theorem (2015)
Let $m$ be a positive integer.
Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and of
discriminant $d_K$ such that $|d_K|\ge
\exp((5+\sqrt{5})(\sqrt{m+1}-1)^2)$.
The associated Dedekind
zeta-function $\zeta_K$ has at most $m$ real zeroes in
$$
\sigma\ge 1-\frac{(5+\sqrt{5})(\sqrt{m+1}-1)^2}{2\log|d_K|}.
$$
Last updated on January 15th, 2024, by Ethan Lee;