Explicit zero-free regions for the \(\zeta\) and \(L\) functions¶
1. Numerical verifications of the Generalized Riemann Hypothesis¶
Numerical verifications of the Riemann hypothesis for the Riemann \(\zeta\)-function have been pushed extremely far. B. Riemann himself computed the first zeros. Concerning more recent published papers, we mention the next result.
Theorem ([Van de Lune et al., 1986])
Every zero \(\rho\) of \(\zeta\) that have a real part between 0 and 1 and an imaginary part not more, in absolute value, than \(\le T_0=545\,439\,823\) are in fact on the critical line, i.e. satisfy \(\Re \rho=1/2\).
The bound \(545\,439\,823\) is increased to \(1\,000\,000\,000\) in [Platt, 2011]. In [Platt, 2017], this bound is further increased to \(30\,610\,046\,000\). Between these results, a group using a network method announced:
Theorem ([Wedeniwski, 2002])
\(T_0=29\,538\,618\,432\) is admissible in the theorem above.
Here is another announcement.
Theorem ([Gourdon and Demichel, 2004])
\(T_0=2.445\cdot 10^{12}\) is admissible in the theorem above.
These two last announcements have not been subject to any academic papers.
Theorem ([Platt and Trudgian, 2021])
\(T_0=3\cdot 10^{12}\) is admissible in the theorem above.
One of the key ingredient is an explicit Riemann-Siegel formula due to [Gabcke, 1979] (the preprint of Gourdon mentionned above gives a version of Gabcke’s result) and such a formula is missing in the case of Dirichlet \(L\)-function.
Let us introduce some terminology. We say that a modulus \(q\ge1\) (i.e. an integer!) satisfies \(GRH(H)\) for some numerical value \(H\) when every zero \(\rho\) of the \(L\)-function associated to a primitive Dirichlet character of conductor \(q\) and whose real part lies within the critical line (i.e. has a real part lying inside the open interval \((0,1)\)) and whose imaginary part is below, in absolute value, \(H\), in fact satisfies \(\Re\rho=1/2\).
The next result was proved by employing an Euler-McLaurin formula.
Theorem ( [Rumely, 1993])
Every \(q\le 13\) satisfies \(GRH(10\,000)\).
Every \(q\) belonging to one of the sets
\(\{k\le 72\}\)
\(\{k\le 112: k\ \text{not prime}\}\)
\(\{116, 117, 120, 121, 124, 125, 128, 132, 140, 143, 144, 156, 163, 169, 180, 216, 243, 256, 360, 420, 432\}\)
satisfies \(GRH(2\,500)\).
These computations have been extended by [Bennett, 2001] by using Rumely’s programm. All these computations have been superseded by the work of D. Platt. who uses two fast Fourier transforms, one in the \(t\)-aspect and one in the \(q\)-aspect, as well as an approximate functionnal equation to prove via extremely rigorous computations that
Theorem ([Platt, 2011], [Platt, 2013])
Every modulus \(q\le 400\,000\) satisfies \(GRH(100\,000\,000/q)\).
We mention here the algorithm of [Omar, 2001] that enables one to prove efficiently that some \(L\)-functions have no zero within the rectangle \(1/2\le \sigma\le1\) et \(2\sigma-|t|=1\) though this algorithm has not been put in practice.
There are much better results concerning real zeros of Dirichlet \(L\)-functions associated to real characters.
2. Asymptotical zero-free regions¶
The first fully explicit zero free region for the Riemann zeta-function is due to [Rosser, 1938] in Lemma 19 (essentially with \(R_0=19\) in the notations below). This is next imporved upon in Theorem 1 of [Rosser and Schoenfeld, 1975] by using a device of [Stechkin, 1970] (getting essentially \(R_0=9.646\)). The next step is in [Ramaré and Rumely, 1996] where the second order term is improved upon, relying on [Stechkin, 1989].
Next, the following result is proven.
Theorem ([Kadiri, 2002], [Kadiri, 2005])
The Riemann \(\zeta\)-function has no zeros in the region \(\displaystyle \Re s \ge 1- \frac1{R_0 \log (| \Im s|+2)}\quad\text{with}\ R_0=5.70175.\)
[Jang and Kwon, 2014] improved the value of \(R_0\) by showing that \(R_0=5.68371\) is admissible. By plugging a better trigonometric polynomial in the same method, it is proved in [Mossinghoff and Trudgian, 2015] that
Theorem (2015)
The Riemann \(\zeta\)-function has no zeros in the region \(\displaystyle \Re s \ge 1- \frac1{R_0 \log (| \Im s|+2)}\quad\text{with}\ R_0=5.573412.\)
Concerning Dirichlet \(L\)-function, the first explicit zero-free region has been obtained in [McCurley, 1984] by adaptating [Rosser and Schoenfeld, 1975]. [Kadiri, 2002] (cf also [Kadiri, 2009]) improves that into:
Theorem (2002)
The Dirichlet \(L\)-functions associated to a character of conductor \(q\) has no zero in the region: \(\displaystyle \Re s \ge 1- \frac1{R_1 \log(q \max(1,| \Im s|))} \quad\text{with}\ R_1=6.4355,\) to the exception of at most one of them which would hence be attached to a real-valued character. This exceptional one would have at most one zero inside the forbidden region (and which is loosely called a “Siegel zero”).
Theorem ([Kadiri, 2018])
The Dirichlet \(L\)-functions associated to a character of conductor \(q\in[3,400\,000]\) has no zero in the region: \(\displaystyle \Re s \ge 1- \frac1{R_2 \log(q \max(1,| \Im s|))} \quad\text{with}\ R_2=5.60.\)
Here is an explicit Vinogradov-Korobov zero-free region.
Theorem ([Ford, 2000])
The Riemann \(\zeta\)-function has no zeros in the region \(\displaystyle \Re s\ge 1-\frac{1}{58(\log |\Im s|)^{2/3}(\log\log |\Im s|)^{1/3}} \quad(|\Im s|\ge 3).\)
A Vinogradov-Korobov zero-free region for Dirchlet \(L\)-functions has later been obtained.
Theorem ([Khale, 2024])
The Riemann \(\zeta\)-function has no zeros in the region \(\displaystyle \Re s\ge 1-\frac{1}{10.5\log q+61.5(\log |\Im s|)^{2/3}(\log\log |\Im s|)^{1/3}} \quad(|\Im s|\ge 10).\)
If we are ready to assume \(|\Im s|\) large enough, the same proof can reduce the 61.5 to 49.13.
Concerning the Dedekind \(\zeta\)-function, see [Kadiri, 2012].
3. Real zeros¶
[Rosser, 1949], [Rosser, 1950], [Chua, 2005], [Watkins, 2004],
Theorem ([Ralaivaosaona and Razakarinoro, 2026])
Let \(d > 3\cdot 10^8\) such that \(−d\) is a fundamental discriminant. The Dirichlet series \(L(s,\chi_d)\) associated to the primitive character :math:chi_d(n)=(frac{-d}{n}) has not zero in the half-plane \(\Re s \ge 1-6.035/\sqrt{d}\).
4. Density estimates¶
After initial work of [Chen and Wang, 1989] and [Liu and Wang, 2002], here are the latest two best results. We first define
\(\displaystyle N(\sigma,T,\chi)=\sum_{\substack{\rho=\beta+i\gamma,\\ L(\rho,\chi)=0,\\ \sigma\le \beta, |\gamma|\le T}}1\) which thus counts the number of zeroes \(\rho\) of \(L(s,\chi)\), zeroes whose real part is denoted by \(\beta\) (and assumed to be larger than \(\sigma\)), and whose imaginary part in absolute value \(\gamma\) is assumed to be not more than \(T\). For the Riemann \(\zeta\)-function (i.e. when \(\chi=\chi_0\) the principal character modulo~1), it is customary to count only the zeroes with positive imaginary part. The relevant number is usually denoted by \(N(\sigma,T)\). We have \(2N(\sigma,T)=N(\sigma,T,\chi_0)\).
For low values, we start with the next theorem. We reproduce only a special case.
Theorem ([Kadiri and Ng, 2012])
Let \(T\ge3.061\cdot10^{10}\). We have \(2N(17/20,T,\chi_0)\le 0.5561T+0.7586\log T-268 658\) where \(\chi_0\) is the principal character modulo 1.
See also [Kadiri, 2013]. Otherwise, here is the result of [Ramaré, 2016].
Theorem (2016)
For \(T\ge2\,000\) and \(T\ge Q\ge10\), as well as \(\sigma\ge0.52\), we have \(\displaystyle \sum_{q\le Q}\frac{q}{\varphi(q)} \sum_{\chi\mod^* q}N(\sigma,T,\chi) \le 20\bigl(56\,Q^{5}T^3\bigr)^{1-\sigma}\log^{5-2\sigma}(Q^2T) +32\,Q^2\log^2(Q^2T)\) where \(\chi\mod^* q\) denotes a sum over all primitive Dirichlet character \(\chi\) to the modulus \(q\). Furthermore, we have \(\displaystyle N(\sigma,T,\chi_0)\le 6T\log T \log\biggl(1+\frac{6.87}{2T}(3T)^{8(1-\sigma)/{3}}\log^{4-2\sigma}(T)\biggr) +103(\log T)^2\) where \(\chi_0\) is the principal character modulo 1.
In [Kadiri et al., 2018]. this result is improved upon, we refer to their paper for their result but quote a corollary.
Theorem (2018)
For \(T\ge1\), we have \(N(0.9,T) \le 1.5\, T^{4/14}\log^{16/5}(T) +3.2\,\log^2 T\) where \(N(\sigma,T)=N(\sigma,T,\chi_0)\) and \(\chi_0\) is the principal character modulo 1.
5. Miscellanae¶
The LMFDBfootnote{url{http://www.lmfdb.org}} database contains the first zeros of many \(L\)-functions. A part of Andrew Odlyzko’s
websitefootnote{url{http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html#GlobalIndex}} contains extensive tables concerning zeroes of the Riemann zeta function.