Bounds on the Dedekind zeta-function¶
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 [Rademacher, 1959] gives the convexity bound. See also section 4.1 of [Trudgian, 2014].
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 \(\displaystyle |\zeta_K(s)|\le 3 \left|\frac{1+s}{1-s}\right|\left(\frac{|d||1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}}\zeta(1+\eta)^n.\)
2. Number of ideals¶
An explicit approximation of the number of ideals in a number field was given in the PhD memoir [Sunley, 1973] of J.S. Sunley. It is recalled in Theorem 1.1 of [Lee, 2023] 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 \(\displaystyle I_K(x) =\kappa_K x +\mathcal{O}^*\biggl(C(K)|\Delta_K|^{\frac{1}{n_K+1}}(\log|\Delta_K|)^{n_K-1} x^{1-\frac{2}{n_K+1}}\biggr)\) where \(\displaystyle C(K)=0.17\frac{6n_K-2}{n_K-1} 2.26^{n_K} e^{4n_K+(26/n_K)}n_K^{n_K+(1/2)} \biggl( 44.39\biggl(\frac{82}{1000}\biggr)^{n_K}n_K! + \frac{13}{n_K-1} \biggr).\)
For \(n_K=2\), 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 [Debaene, 2019].
Theorem (2019)
The notation being as above, we have \(\displaystyle I_K(x)=\kappa_K x +O^*\biggl(n_K^{10n_K^2} (\text{Reg}_Kh_K)^{1/n_K}\biggr) (1+\log\text{Reg}_Kh_K)^{\frac{(n_K-1)^2}{n_K}} x^{1-\frac{1}{n_K}}.\)
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 [Gun et al., 2023].
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 [Louboutin, 2000] and in Section 3 of [Garcia and Lee, 2022].
Theorem (2000, 2022)
If \(n_K\geq 3\), then \(\displaystyle \frac{0.0014480}{n_K g(n_K){|\Delta_K|}^{1/n_K}} < \kappa_K \leq \left(\frac{e\log |\Delta_K| }{2(n_K - 1)}\right)^{n_K - 1},\) in which \(g(n_K)=1\) 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 [Garcia and Lee, 2022].
Theorem (2022)
Assume that the Generalised Riemann Hypothesis and the Dedekind Conjecture are true. If \(n_K\geq 2\), then \(\displaystyle \frac{e^{-17.81(n_K -1)}}{\log\log{|\Delta_K|}} \leq \kappa_{K} \leq e^{17.81(n_K -1 )} (\log\log{|\Delta_K|})^{n_K-1}.\)
4. Zeroes and zero-free regions¶
We denote by \(N_K(T)\) 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 such that \(|\Im \rho|\le T\). After a first result in [Kadiri and Ng, 2012], we find in [Trudgian, 2015] the following result.
Theorem (2014)
When \(T\ge1\), we have \(\displaystyle N_K(T)=\frac{T}{\pi}\log\Bigl(|d_K|\Big(\frac{T}{2\pi e}\Bigr)^n\Bigr) +O^*\bigl(0.316(\log |d_K|+n\log T)+5.872 n+3.655\bigr)\).
This is improved in [Hasanalizade et al., 2021] into:
Theorem (2021)
When \(T\ge 1\), 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 [Kadiri, 2012], 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 \(\displaystyle \sigma\ge 1-\frac{1}{12.55\log|d_K|+n(9.69\log|t|+3.03)+58.63}, |t|\ge1\) and at most one zero in the region \(\displaystyle \sigma\ge 1-\frac{1}{12.74\log|d_K|}, |t|\le 1.\) The exceptional zero, if it exists, is simple and real.
This is improved in [Lee, 2021] 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 \(\displaystyle \sigma\ge 1-\frac{1}{12.2411\log|d_K|+n(9.5347\log|t|+0.0501)+2.2692}, |t|\ge1\) and if \(d_K\) is sufficiently large, then there is at most one zero in the region \(\displaystyle \sigma\ge 1-\frac{1}{12.4343\log|d_K|}, |t|< 1.\) The exceptional zero, if it exists, is simple and real.
See [Ahn and Kwon, 2014] for a result for Hecke \(L\)-series.
In [Louboutin, 2017], 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 \(\displaystyle \sigma\ge 1-\frac{1}{1.7\log|d_K|}, |t|\ge\frac{1}{4\log|d_K|},\) \(\displaystyle \sigma\ge 1-\frac{1}{2\log|d_K|}, |t|\ge\frac{1}{2\log|d_K|},\) and \(\displaystyle \sigma\ge 1-\frac{1}{1.62\log|d_K|}, t=0.\)
5. Real Zeroes.¶
In [Louboutin, 2015], 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 \(\displaystyle \sigma\ge 1-\frac{(5+\sqrt{5})(\sqrt{m+1}-1)^2}{2\log|d_K|}.\)