Bounds for :math:`|\zeta(s)|`, :math:`|L(s,\chi)|` and related questions ======================================================================== .. if-builder:: html .. toctree:: :maxdepth: 2 Collecting references: :cite:`Trudgian11`, :cite:`Kadiri-Ng12`. 1. Approximating :math:`|\zeta(s)|` or :math:`L`-series in the critical strip ----------------------------------------------------------------------------- :cite:`Reyna11` extends the Phd memoir :cite:`Gabcke79` and provides an explicit Riemann-Siegel formula for :math:`\zeta(s)`. Theorem 1.2 of :cite:`Kadiri13` proves the following. .. admonition:: Theorem (2013) :class: thm-tme-emt When :math:`t > t_0 > 0`, :math:`c > 1/(2\pi)` and :math:`s = \sigma +it` with :math:`\sigma\ge 1/2`, we have :math:`\displaystyle \zeta(s) =\sum_{n < c t} \frac{1}{n^s} + \mathcal{O}^* \biggl( \biggl(c+\tfrac12+\frac{3\sqrt{1+1/t_0^2}}{2\pi} \biggl(\frac{\pi}{12c}+1+\frac{1}{2\pi c-1}\biggr) \biggr) (ct)^{-\sigma}\biggr).` Notice that, by using the constant :math:`c`, we may deduce from this an approximation of :math:`\zeta(s)` by a fixed Dirichlet polynomial when :math:`T\le t\le 2T`, for some parameter :math:`T`. 2. Size of :math:`|\zeta(s)|` and of :math:`L`-series ----------------------------------------------------- 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.` On the line :math:`\Re s=1/2`, Lemma 2 of :cite:`Lehman70` gives a better result, namely .. admonition:: Theorem (1970) :class: thm-tme-emt If :math:`t\ge 1/5`, we have :math:`|\zeta(\tfrac12+it)|\le 4 (t/(2\pi))^{1/4}`. In fact, Lehman states this Theorem for :math:`t\ge 64/(2\pi)`, but modern means of computations makes it easy to check that it holds as soon as :math:`t\ge 0.2`. See also equation (56) of :cite:`Backlund18` reproduced below. For Dirichlet :math:`L`-series, Theorem 3 of :cite:`Rademacher59` gives the corresponding convexity bound. .. admonition:: Theorem (1959) :class: thm-tme-emt In the strip :math:`-\eta\le \sigma\le 1+\eta`, :math:`0 < \eta\le 1/2`, for all moduli :math:`q > 1` and all primitive characters :math:`\chi` modulo :math:`q`, the inequality :math:`\displaystyle |L(s,\chi)|\le \left(q\frac{|1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}}\zeta(1+\eta)` holds. This paper contains other similar convexity bounds. Corollary to Theorem 3 of :cite:`Cheng-Graham01` goes beyond convexity. .. admonition:: Theorem (2001) :class: thm-tme-emt For :math:`0\le t\le e`, we have :math:`|\zeta(\tfrac12+it)|\le 2.657`. For :math:`t\ge e`, we have :math:`|\zeta(\tfrac12+it)|\le 3t^{1/6}\log t`. Section 5 of bibref("Trudgian*13") shows that one can replace the constant 3 by 2.38. This is improved in :cite:`Hiary16`. .. admonition:: Theorem (2016) :class: thm-tme-emt When :math:`t\ge 3`, we have :math:`|\zeta(\tfrac12+it)|\le 0.63t^{1/6}\log t`. Concerning :math:`L`-series, the situation is more difficult but :cite:`Hiary16b` manages, among other and more precise results, to prove the following. .. admonition:: Theorem (2016) :class: thm-tme-emt Assume :math:`\chi` is a primitive Dirichlet character modulo :math:`q>1`. Assume further that :math:`q` is a sixth power. Then, when :math:`|t|\ge 200`, we have :math:`\displaystyle |L(\tfrac12+it,\chi)|\le 9.05d(q) (q|t|)^{1/6}(\log q|t|)^{3/2}` where :math:`d(q)` is the number of divisors of :math:`q`. It is often useful to have a representation of the Riemann zeta function or of :math:`L`-series inside the critical strip. In the case of :math:`L`-series, :cite:`Spira69` and :cite:`Rumely93` proceed via decomposition in Hurwitz zeta function which they compute through an Euler-MacLaurin development. We have a more efficient approximation of the Riemann zeta function provided by the Riemann Siegel formula, see for instance equations (3-2)--(3.3) of :cite:`Odlyzko87`. This expression is due to :cite:`Gabcke79`. See also equations (2.4)-(2.5) of :cite:`Lehman66`, a corrected version of Theorem 2 of :cite:`Titchmarsh47`. In general, we have the following estimate taken from equations (53)-(54), (56) and (76) of :cite:`Backlund18` (see also :cite:`Backlund14`). .. admonition:: Theorem (1918) :class: thm-tme-emt * When :math:`t\ge 50` and :math:`\sigma\ge1`, we have :math:`|\zeta(\sigma+it)|\le \log t-0.048`. * When :math:`t\ge 50` and :math:`0\le \sigma\le1`, we have :math:`|\zeta(\sigma+it)|\le\frac{t^2}{t^2-4}\left(\frac{t}{2\pi}\right)^{\frac{1-\sigma}{2}}\log t`. * When :math:`t\ge 50` and :math:`-1/2\le \sigma\le0`, we have :math:`|\zeta(\sigma+it)|\le\left(\frac{t}{2\pi}\right)^{\frac{1}{2}-\sigma}\log t`. On the line :math:`\Re s=1`, :cite:`Trudgian12b` establishes the next result. .. admonition:: Theorem (2012) :class: thm-tme-emt When :math:`t\ge 3`, we have :math:`|\zeta(1+it)|\le\tfrac34 \log t`. The paper :cite:`Patel22` proves the next bound. .. admonition:: Theorem (2022) :class: thm-tme-emt When :math:`t\ge 3`, we have :math:`|\zeta(1+it)|\le\min\bigl(\frac34 \log t, \frac12\log t + 1.93, \frac15 \log t + 44.02\bigr)`. Asymptotically better bounds are available since the work of :cite:`Ford02`. .. admonition:: Theorem (2002) :class: thm-tme-emt When :math:`t\ge 3` and :math:`1/2\le \sigma\le 1`, we have :math:`|\zeta(\sigma+it)|\le 76.2 t^{4.45(1-\sigma)^{3/2} } (\log t)^{2/3}`. The constants are still too large for this result to be of use in any decent region. See :cite:`Kulas94` for an earlier estimate. 3. On the total number of zeroes -------------------------------- The first explicit estimate for the number of zeros of the Riemann :math:`\zeta`-function goes back to :cite:`Backlund14`. An elegant consequence of the result of Backlund is the following easy estimate taken from Lemma 1 of :cite:`Lehman66a`. .. admonition:: Theorem (1966) :class: thm-tme-emt If :math:`\varphi` is a continuous function which is positive and monotone decreasing for :math:`2\pi e\le T_1\le t\le T_2`, then :math:`\displaystyle \sum_{T_1 < \gamma\le T_2} \varphi(\gamma) =\frac{1}{2\pi}\int_{T_1}^{T_2}\varphi(t)\log\frac{t}{2\pi}dt +O^*\biggl(4\varphi(T_1)\log T_1+2\int_{T_1}^{T_2}\frac{\varphi(t)}{t} dt\biggr)` where the summation is over all zeros of the Riemann :math:`\zeta`-function of imaginary part between :math:`T_1` and :math:`T_2`, with multiplicity. Theorem 19 of :cite:`Rosser41` gives a bound for the total number of zeroes. .. admonition:: Theorem (1941) :class: thm-tme-emt For :math:`T\ge2`, we have :math:`\displaystyle N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\Bigl(0.137\log T+0.443\log\log T+1.588 \Bigr)` where the summation is over all zeros of the Riemann :math:`\zeta`-function of imaginary part between 0 and :math:`T`, with multiplicity. It is noted in Lemma 1 of :cite:`Ramare-Saouter02` that the :math:`O`-term can be replaced by the simpler :math:`O^*(0.67\log\frac{T}{2\pi})` when :math:`T\ge 10^3`. This is improved in Corollary 1 of :cite:`Trudgian13` into .. admonition:: Theorem (2014) :class: thm-tme-emt For :math:`T\ge e`, we have :math:`\displaystyle N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\bigl(0.112\log T+0.278\log\log T+2.510+\frac{1}{5T} \bigr)` where the summation is over all zeros of the Riemann :math:`\zeta`-function of imaginary part between 0 and :math:`T`, with multiplicity. Corollary 1.4 of the main theorem of :cite:`Hasanalizade-Shen-Wong22` reads .. admonition:: Theorem (2022) :class: thm-tme-emt For :math:`T\ge e`, we have :math:`\displaystyle N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}}1=\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8}+O^*\bigl(0.1038\log T+0.2573\log\log T+9.3675\bigr)` where the summation is over all zeros of the Riemann :math:`\zeta`-function of imaginary part between 0 and :math:`T`, with multiplicity. We may also replace :math:`0.1038\log T+0.2573\log\log T+9.3675` by :math:`0.1095\log T+0.2042\log\log T+3.0305`. Concerning Dirichlet :math:`L`-functions, the paper :cite:`Bennett-Martin-OBryant-Rechnitzer21` contains the next result. .. admonition:: Theorem (2021) :class: thm-tme-emt Let :math:`\chi` be a Dirichlet character of conductor :math:`q > 1`. For :math:`T\ge 5/7` and :math:`\ell= \log\frac{q(T+2)}{2\pi} > 1.567`, we have :math:`\displaystyle N(T,\chi)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1=\frac{T}{\pi}\log\frac{qT}{2\pi}-\frac{T}{\pi}+\frac{\chi(-1)}{4}+O^*\bigl(0.22737\ell+2\log(1+\ell)-0.5 \bigr)` where the summation is over all zeros of the Dirichlet function :math:`L(\cdot,\chi)` of imaginary part between :math:`-T` and :math:`T`, with multiplicity. 4. :math:`L^2`-averages ----------------------- In Theorem 1.4 of :cite:`Kadiri13`, we find the next result. .. admonition:: Theorem (2013) :class: thm-tme-emt When :math:`0.5208 < \sigma < 0.9723` and :math:`10^3\le H \le 10^{10}`, we have, for any :math:`T\ge H`, :math:`\displaystyle \int_H^T |\zeta(\sigma + t)|^2 dt\le (T-H) \bigl(\zeta(2\sigma) +\mathcal{E}_1(\sigma, H)\bigr)` where :math:`\mathcal{E}_1(\sigma, H)` is a small error term whose precise expression in given the stated paper. We can find in :cite:`Helfgott17u` the proof of the following estimate. Though it is unpublished yet, the full proof is available. .. admonition:: Theorem (2019) :class: thm-tme-emt Let :math:`0 < \sigma\le1` and :math:`T \ge 3`. Then :math:`\displaystyle \frac{1}{2\pi}\biggl(\int_{\sigma-i\infty}^{\sigma-iT}+ \int^{\sigma+i\infty}_{\sigma+iT} \biggr) \frac{|\zeta(s)|^2}{|s|^2}ds\le\kappa_{\sigma,T} \begin{cases} \frac{c_{1,\sigma}}{T}+\frac{c^\flat_{1,\sigma}}{T^{2\sigma}} &\text{when }\sigma > 1/2,\\ \frac{\log T}{2T}+\frac{c^\flat_{2,\sigma}}{T} &\text{when }\sigma=1/2,\\ c_{3,\sigma}/T^{2\sigma}&\text{when }\sigma < 1/2. \end{cases}` where :math:`\displaystyle c_{1,\sigma}=\zeta(2\sigma)/2, c_{1,\sigma}^\flat=c^2 \frac{3^{2\sigma}}{2\sigma}, c_{2,\sigma}^\flat=3c^2+\frac{1-\log 3}{2},c=9/16` :math:`\displaystyle c_{3,\sigma}=\Bigl(\frac{c^2}{2\sigma}+\frac{1/6}{1-2\sigma}\Bigr)\Bigl(1+\frac{1}{\sigma}\Bigr)^{2\sigma},\kappa_{\sigma,T}=\begin{cases}\frac{9/4}{\left(1-\frac{9/2}{T^2}\right)^2}&\text{when }1/2\le \sigma\le 1,\\\frac{(1+\sigma)^2}{\left(1-\frac{(1+\sigma)^2}{\sigma T^2}\right)^2}&\text{when }0 < \sigma < 1/2. \end{cases}` 5. Bounds on the real line -------------------------- After some estimates from :cite:`Bastion-Rogalaski02` Lemma 5.1 of :cite:`Ramare13d` shows the following. .. admonition:: Theorem (2013) :class: thm-tme-emt When :math:`\sigma> 1` and :math:`t` is any real number, we have :math:`|\zeta(\sigma+it)|\le e^{\gamma(\sigma-1) }/(\sigma-1)`. Here is the Theorem of :cite:`Delange87`. See also Lemma 2.3 of :cite:`Ford01` for a slightly weaker version. .. admonition:: Theorem (1987) :class: thm-tme-emt When :math:`\sigma> 1` and :math:`t` is any real number, we have :math:`\displaystyle -\Re\frac{\zeta'}{\zeta}(\sigma+it)\le \frac{1}{\sigma-1}-\frac{1}{2\sigma^2}.`