Explicit zero-free regions for the :math:`\zeta` and :math:`L` functions ======================================================================== .. if-builder:: html .. toctree:: :maxdepth: 2 1. Numerical verifications of the Generalized Riemann Hypothesis ---------------------------------------------------------------- Numerical verifications of the Riemann hypothesis for the Riemann :math:`\zeta`-function have been pushed extremely far. B. Riemann himself computed the first zeros. Concerning more recent published papers, we mention the next result. .. admonition:: Theorem (:cite:`Lune-Riele-Winter86`) :class: thm-tme-emt Every zero :math:`\rho` of :math:`\zeta` that have a real part between 0 and 1 and an imaginary part not more, in absolute value, than :math:`\le T_0=545\,439\,823` are in fact on the critical line, i.e. satisfy :math:`\Re \rho=1/2`. The bound :math:`545\,439\,823` is increased to :math:`1\,000\,000\,000` in :cite:`Platt11`. In :cite:`Platt17`, this bound is further increased to :math:`30\,610\,046\,000`. Between these results, a group using a network method announced: .. admonition:: Theorem (:cite:`Wedeniwski02`) :class: thm-tme-emt :math:`T_0=29\,538\,618\,432` is admissible in the theorem above. Here is another announcement. .. admonition:: Theorem (:cite:`Gourdon-Demichel04`) :class: thm-tme-emt :math:`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. .. admonition:: Theorem (:cite:`Platt-Trudgian21a`) :class: thm-tme-emt :math:`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 :cite:`Gabcke79` (the preprint of Gourdon mentionned above gives a version of Gabcke's result) and such a formula is missing in the case of Dirichlet :math:`L`-function. Let us introduce some terminology. We say that a modulus :math:`q\ge1` (i.e. an integer!) satisfies :math:`GRH(H)` for some numerical value :math:`H` when every zero :math:`\rho` of the :math:`L`-function associated to a primitive Dirichlet character of conductor :math:`q` and whose real part lies within the critical line (i.e. has a real part lying inside the open interval :math:`(0,1)`) and whose imaginary part is below, in absolute value, :math:`H`, in fact satisfies :math:`\Re\rho=1/2`. The next result was proved by employing an Euler-McLaurin formula. .. admonition:: Theorem ( :cite:`Rumely93`) :class: thm-tme-emt * Every :math:`q\le 13` satisfies :math:`GRH(10\,000)`. * Every :math:`q` belonging to one of the sets * :math:`\{k\le 72\}` * :math:`\{k\le 112: k\ \text{not prime}\}` * :math:`\{116, 117, 120, 121, 124, 125, 128, 132, 140, 143, 144, 156, 163, 169, 180, 216, 243, 256, 360, 420, 432\}` satisfies :math:`GRH(2\,500)`. These computations have been extended by :cite:`Bennett01` 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 :math:`t`-aspect and one in the :math:`q`-aspect, as well as an approximate functionnal equation to prove via extremely rigorous computations that .. admonition:: Theorem (:cite:`Platt11`, :cite:`Platt13`) :class: thm-tme-emt Every modulus :math:`q\le 400\,000` satisfies :math:`GRH(100\,000\,000/q)`. We mention here the algorithm of :cite:`Omar01` that enables one to prove efficiently that some :math:`L`-functions have no zero within the rectangle :math:`1/2\le \sigma\le1` et :math:`2\sigma-|t|=1` though this algorithm has not been put in practice. There are much better results concerning real zeros of Dirichlet :math:`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 :cite:`Rosser38` in Lemma 19 (essentially with :math:`R_0=19` in the notations below). This is next imporved upon in Theorem 1 of :cite:`Rosser-Schoenfeld75` by using a device of :cite:`Stechkin70` (getting essentially :math:`R_0=9.646`). The next step is in :cite:`Ramare-Rumely96` where the second order term is improved upon, relying on :cite:`Stechkin89`. Next, the following result is proven. .. admonition:: Theorem (:cite:`Kadiri02`, :cite:`Kadiri05`) :class: thm-tme-emt The Riemann :math:`\zeta`-function has no zeros in the region :math:`\displaystyle \Re s \ge 1- \frac1{R_0 \log (| \Im s|+2)}\quad\text{with}\ R_0=5.70175.` :cite:`Jang-Kwon14` improved the value of :math:`R_0` by showing that :math:`R_0=5.68371` is admissible. By plugging a better trigonometric polynomial in the same method, it is proved in :cite:`Mossinghoff-Trudgian15` that .. admonition:: Theorem (2015) :class: thm-tme-emt The Riemann :math:`\zeta`-function has no zeros in the region :math:`\displaystyle \Re s \ge 1- \frac1{R_0 \log (| \Im s|+2)}\quad\text{with}\ R_0=5.573412.` Concerning Dirichlet :math:`L`-function, the first explicit zero-free region has been obtained in :cite:`McCurley84-1` by adaptating :cite:`Rosser-Schoenfeld75`. :cite:`Kadiri02` (cf also :cite:`Kadiri02-2`) improves that into: .. admonition:: Theorem (2002) :class: thm-tme-emt The Dirichlet :math:`L`-functions associated to a character of conductor :math:`q` has no zero in the region: :math:`\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"). .. admonition:: Theorem (:cite:`Kadiri18`) :class: thm-tme-emt The Dirichlet :math:`L`-functions associated to a character of conductor :math:`q\in[3,400\,000]` has no zero in the region: :math:`\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. .. admonition:: Theorem (:cite:`Ford01`) :class: thm-tme-emt The Riemann :math:`\zeta`-function has no zeros in the region :math:`\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 :math:`L`-functions has later been obtained. .. admonition:: Theorem (:cite:`Khale24`) :class: thm-tme-emt The Riemann :math:`\zeta`-function has no zeros in the region :math:`\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 :math:`|\Im s|` large enough, the same proof can reduce the 61.5 to 49.13. Concerning the Dedekind :math:`\zeta`-function, see :cite:`Kadiri12`. 3. Real zeros -------------- :cite:`Rosser49`, :cite:`Rosser50`, :cite:`Chua05`, :cite:`Watkins00-1`, .. admonition:: Theorem (:cite:`Ralaivaosaona-Razakarinoro26`) :class: thm-tme-emt Let :math:`d > 3\cdot 10^8` such that :math:`−d` is a fundamental discriminant. The Dirichlet series :math:`L(s,\chi_d)` associated to the primitive character :\math:`\chi_d(n)=(\frac{-d}{n})` has not zero in the half-plane :math:`\Re s \ge 1-6.035/\sqrt{d}`. 4. Density estimates -------------------- After initial work of :cite:`Chen-Wang89-2` and :cite:`Liu-Wang02-1`, here are the latest two best results. We first define :math:`\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 :math:`\rho` of :math:`L(s,\chi)`, zeroes whose real part is denoted by :math:`\beta` (and assumed to be larger than :math:`\sigma`), and whose imaginary part in absolute value :math:`\gamma` is assumed to be not more than :math:`T`. For the Riemann :math:`\zeta`-function (i.e. when :math:`\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 :math:`N(\sigma,T)`. We have :math:`2N(\sigma,T)=N(\sigma,T,\chi_0)`. For low values, we start with the next theorem. We reproduce only a special case. .. admonition:: Theorem (:cite:`Kadiri-Ng12`) :class: thm-tme-emt Let :math:`T\ge3.061\cdot10^{10}`. We have :math:`2N(17/20,T,\chi_0)\le 0.5561T+0.7586\log T-268 658` where :math:`\chi_0` is the principal character modulo 1. See also :cite:`Kadiri13`. Otherwise, here is the result of :cite:`Ramare13d`. .. admonition:: Theorem (2016) :class: thm-tme-emt For :math:`T\ge2\,000` and :math:`T\ge Q\ge10`, as well as :math:`\sigma\ge0.52`, we have :math:`\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 :math:`\chi\mod^* q` denotes a sum over all primitive Dirichlet character :math:`\chi` to the modulus :math:`q`. Furthermore, we have :math:`\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 :math:`\chi_0` is the principal character modulo 1. In :cite:`Kadiri-Lumley-Ng18`. this result is improved upon, we refer to their paper for their result but quote a corollary. .. admonition:: Theorem (2018) :class: thm-tme-emt For :math:`T\ge1`, we have :math:`N(0.9,T) \le 1.5\, T^{4/14}\log^{16/5}(T) +3.2\,\log^2 T` where :math:`N(\sigma,T)=N(\sigma,T,\chi_0)` and :math:`\chi_0` is the principal character modulo 1. 5. Miscellanae ---------------- The LMFDB\footnote{\url{http://www.lmfdb.org}} database contains the first zeros of many :math:`L`-functions. A part of Andrew Odlyzko's website\footnote{\url{http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html#GlobalIndex}} contains extensive tables concerning zeroes of the Riemann zeta function.