Bounds on the Dedekind zeta-function
====================================

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1. Size 
--------


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 :cite:`Rademacher59` gives
the convexity bound. See also section 4.1 of
:cite:`Trudgian13`.

 
.. admonition:: Theorem (1959)
   :class: thm-tme-emt

   In the strip :math:`-\eta\le \sigma\le 1+\eta`, :math:`0 < \eta\le 1/2`, the Dedekind zeta
   function :math:`\zeta_K(s)` belonging to the algebraic number field :math:`K` of degree
   :math:`n` and discriminant :math:`d` satisfies the inequality
   :math:`\displaystyle  |\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.` 




 

2. Number of ideals
-------------------


An explicit approximation of the number of ideals in a number field
was given in the PhD memoir
:cite:`Sunley73`
of J.S. Sunley. It is recalled in Theorem 1.1 of
:cite:`Lee22`
and further refined there in Theorem 1.2.

 
.. admonition:: Theorem (2022)
   :class: thm-tme-emt

   For :math:`x > 0` and :math:`n_K\ge 0`, the number :math:`I_K(x)` of integral ideals
   of norm at most :math:`x` in the
   number field :math:`K` of degree :math:`n_K` and discriminant :math:`\Delta_K` is approximated given by
   :math:`\displaystyle      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
   :math:`\displaystyle      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 :math:`n_K=2`, the constant :math:`\Delta_K` is about :math:`8.80\cdot 10^{11}`. The
constant arising from Sunley's work was about :math:`1.75\cdot 10^{30}`.

 
For :math:`n_K=3`, the constant :math:`C(K)` is approximately equal to :math:`8.45\cdot 10^{11}`. The
constant arising from Sunley's work was about :math:`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
:cite:`Debaene19`.

 
.. admonition:: Theorem (2019)
   :class: thm-tme-emt

   The notation being as above, we have
   :math:`\displaystyle    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
:cite:`Gun-Ramare-Sivaraman22b`.

3. Bounds for the residue of the Dedekind zeta-function
-------------------------------------------------------

Let :math:`K` be a number field over :math:`\mathbb{Q}` with degree :math:`n_K` and
discriminant :math:`\Delta_K`. Furthermore, suppose that the residue of the
Dedekind zeta function :math:`\zeta_K(s)` at :math:`s=1` is denoted
:math:`\kappa_K`. Unconditional bounds for the residue of the Dedekind
zeta-function at :math:`s=1` are found in
:cite:`Louboutin00` and in
Section 3 of
:cite:`Garcia-Lee22`.

 
.. admonition:: Theorem (2000, 2022)
   :class: thm-tme-emt

   If :math:`n_K\geq 3`, then
   :math:`\displaystyle      \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 :math:`g(n_K)=1` if :math:`K` has a normal tower over :math:`\mathbb{Q}` and
   :math:`g(n_K) = n_K!` otherwise.



If the Generalised Riemann Hypothesis and Dedekind Conjecture
(i.e. :math:`\zeta_K/\zeta` is entire) are
true, then stronger bounds are found in Corollary 2 of
:cite:`Garcia-Lee22b`.

 
.. admonition:: Theorem (2022)
   :class: thm-tme-emt

   Assume that the Generalised Riemann Hypothesis and the Dedekind
   Conjecture are true. If :math:`n_K\geq 2`, then 
   :math:`\displaystyle      \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 :math:`N_K(T)` the number of zeros :math:`\rho`, of the Dedekind
zeta-function of the number field :math:`K` of degree :math:`n` and discriminant
:math:`d_K`,
zeros that lie in the critical strip
:math:`0 < \Re \rho = \sigma < 1` and such that :math:`|\Im \rho|\le T`.
After a first result in
:cite:`Kadiri-Ng12`,
we find in
:cite:`Trudgian14-1`
the following result.


 
.. admonition:: Theorem (2014)
   :class: thm-tme-emt

   When :math:`T\ge1`, we have
   :math:`\displaystyle 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 
:cite:`Hasanalizade-Shen-Wong21`
into:


 
.. admonition:: Theorem (2021)
   :class: thm-tme-emt

   When :math:`T\ge 1`, we have
   :math:`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
:cite:`Kadiri12`,
a zero-free region is proved.


 
.. admonition:: Theorem (2012)
   :class: thm-tme-emt

   Let :math:`K` be a number field of degree :math:`n` over :math:`\mathbb{Q}` and of
   discriminant :math:`d_K` such that :math:`|d_K| \ge 2`. The associated Dedekind
   zeta-function :math:`\zeta_K` has no zeros in the region
   :math:`\displaystyle  \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
   :math:`\displaystyle  \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
:cite:`Lee21a`
into:

 
.. admonition:: Theorem (2021)
   :class: thm-tme-emt

   Let :math:`K` be a number field of degree :math:`n` over :math:`\mathbb{Q}` and of
   discriminant :math:`d_K` such that :math:`|d_K| \ge 2`. The associated Dedekind
   zeta-function :math:`\zeta_K` has no zeros in the region
   :math:`\displaystyle  \sigma\ge 1-\frac{1}{12.2411\log|d_K|+n(9.5347\log|t|+0.0501)+2.2692}, |t|\ge1` 
   and if :math:`d_K` is sufficiently large, then there is at most one zero in the region
   :math:`\displaystyle  \sigma\ge 1-\frac{1}{12.4343\log|d_K|}, |t|< 1.` 
   The exceptional zero, if it exists, is simple and real.



See
:cite:`Ahn-Kwon14`
for a result for Hecke :math:`L`-series.

In
:cite:`Louboutin17`,
a zero-free region is proved. Here is slightly simplified version of
his result.


 
.. admonition:: Theorem (2017)
   :class: thm-tme-emt

   Let :math:`K` be a number field of degree :math:`n` over :math:`\mathbb{Q}` and of
   discriminant :math:`d_K` such that :math:`|d_K| \ge 8`. The associated Dedekind
   zeta-function :math:`\zeta_K` has no zeros in the regions
   :math:`\displaystyle  \sigma\ge 1-\frac{1}{1.7\log|d_K|}, |t|\ge\frac{1}{4\log|d_K|},` 
   :math:`\displaystyle  \sigma\ge 1-\frac{1}{2\log|d_K|}, |t|\ge\frac{1}{2\log|d_K|},` 
   and
   :math:`\displaystyle  \sigma\ge 1-\frac{1}{1.62\log|d_K|}, t=0.` 




5. Real Zeroes.
----------------


In
:cite:`Louboutin15b`,
we find the next result.


 
.. admonition:: Theorem (2015)
   :class: thm-tme-emt

   Let :math:`m` be a positive integer.
   Let :math:`K` be a number field of degree :math:`n` over :math:`\mathbb{Q}` and of
   discriminant :math:`d_K` such that
   :math:`|d_K|\ge \exp((5+\sqrt{5})(\sqrt{m+1}-1)^2)`
   The associated Dedekind
   zeta-function :math:`\zeta_K` has at most :math:`m` real zeroes in
   :math:`\displaystyle  \sigma\ge 1-\frac{(5+\sqrt{5})(\sqrt{m+1}-1)^2}{2\log|d_K|}.`