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 who proved that
Theorem (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 . In , this bound is further increased to $30\,610\,046\,000$. Between these results, announced that, he and many collaborators proved, using a network method:
Theorem (2002)
$T_0=29\,538\,618\,432$ is admissible in the theorem above.
went one step further
Theorem (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. We now have
Theorem (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 (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 $\Re\rho=1/2$. By employing an Euler-McLaurin formula, has proved that
Theorem (1993)
  • Every $q\le 13$ satisfies $GRH(10\,000)$.
  • Every $q$ belonging to one of the sets
    •   $\{k\le 72\}$
    •   $\{k\le 112, \text{$k$ non premier}\}$
    •  $\begin{aligned}\{116, 117, &120, 121, 124, 125, 128, 132, 140, 143, 144, 156, 163, \\ &169, 180, 216, 243, 256, 360, 420, 432\}\end{aligned}$
    satisfies $GRH(2\,500)$.
These computations have been extended by by using Rumely's programm. All these computations have been superseded by the work of D. Platt. and use 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 (2011-2013)
Every modulus $q\le 400\,000$ satisfies $GRH(100\,000\,000/q)$.
We mention here the algorithm of 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 in Lemma 19 (essentially with $R_0=19$ in the notations below). This is next imporved upon in Theorem 1 of by using a device of (getting essentially $R_0=9.646$). The next step is in where the second order term is improved upon, relying on . Next, in and later in , the following result is proven.
Theorem (2002)
The Riemann $\zeta$-function has no zeros in the region $$ \Re s \ge 1- \frac1{R_0 \log (| \Im s|+2)}\quad\text{with}\ R_0=5.70175. $$
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 that
Theorem (2015)
The Riemann $\zeta$-function has no zeros in the region $$ \Re s \ge 1- \frac1{R_0 \log (| \Im s|+2)}\quad\text{with}\ R_0=5.573412. $$
Concerning Dirichlet $L$-function, the first explicit zero-free region has been obtained in by adaptating . (cf also ) improves that into:
Theorem (2002)
The Dirichlet $L$-functions associated to a character of conductor $q$ has no zero in the region: $$ \Re s \ge 1- \frac1{R_1 \log(q \max(1,| \Im s|))} \quad\text{with}\ R_1=6.4355, $$ 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").
In , the next theorem is proved.
Theorem (2016)
The Dirichlet $L$-functions associated to a character of conductor $q\in[3,400\,000]$ has no zero in the region: $$ \Re s \ge 1- \frac1{R_2 \log(q \max(1,| \Im s|))} \quad\text{with}\ R_1=5.60. $$
Concerning the Vinogradov-Korobov zero-free region, shows that
Theorem (2001)
The Riemann $\zeta$-function has no zeros in the region $$ \Re s\ge 1-\frac{1}{58(\log |\Im s|)^{2/3}(\log\log |\Im s|)^{1/3}} \quad(|\Im s|\ge 3). $$
Concerning the Dedekind $\zeta$-function, see .
3. Real zeros
, , , ,
4. Density estimates
After initial work of and , here are the latest two best results. We first define $$ N(\sigma,T,\chi)=\sum_{\substack{\rho=\beta+i\gamma,\\ L(\rho,\chi)=0,\\ \sigma\le \beta, |\gamma|\le T}}1 $$ 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 main Theorem of . We reproduce only a special case.
Theorem (2013)
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 . Otherwise, here is the result of .
Theorem (2016)
For $T\ge2\,000$ and $T\ge Q\ge10$, as well as $\sigma\ge0.52$, we have $$ \sum_{q\le Q}\frac{q}{\varphi(q)} \sum_{\chi\mod^* q}N(\sigma,T,\chi) \le 20\bigl(56\,Q^{5}T^3\bigr)^{1-\sigma}\log^{5-2\sigma}(Q^2T) +32\,Q^2\log^2(Q^2T) $$ where $\chi\mod^* q$ denotes a sum over all primitive Dirichlet character $\chi$ to the modulus $q$. Furthermore, we have $$ N(\sigma,T,\chi_0)\le 6T\log T \log\biggl(1+\frac{6.87}{2T}(3T)^{8(1-\sigma)/{3}}\log^{4-2\sigma}(T)\biggr) +103(\log T)^2 $$ where $\chi_0$ is the principal character modulo 1.
In . this result is improved upon, we refer to their paper for their result by quote a corollary.
For $T\ge1$, we have $ N(0.9,T) \le 11.5\, 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 LMFDB database contains the first zeros of many $L$-functions. A part of Andrew Odlyzko's website contains extensive tables concerning zeroes of the Riemann zeta function.
Last updated on September 19th, 2021, by Olivier Ramaré