Bounds for \(|\zeta(s)|\), \(|L(s,\chi)|\) and related questions

Collecting references: [Trudgian, 2011], [Kadiri and Ng, 2012].

1. Approximating \(|\zeta(s)|\) or \(L\)-series in the critical strip

[Arias de Reyna, 2011] extends the Phd memoir [Gabcke, 1979] and provides an explicit Riemann-Siegel formula for \(\zeta(s)\).

Theorem 1.2 of [Kadiri, 2013] proves the following.

Theorem (2013)

When \(t > t_0 > 0\), \(c > 1/(2\pi)\) and \(s = \sigma +it\) with \(\sigma\ge 1/2\), we have \(\displaystyle \zeta(s) =\sum_{n < c t} \frac{1}{n^s} + \mathcal{O}^* \biggl( \biggl(c+\tfrac12+\frac{3\sqrt{1+1/t_0^2}}{2\pi} \biggl(\frac{\pi}{12c}+1+\frac{1}{2\pi c-1}\biggr) \biggr) (ct)^{-\sigma}\biggr).\)

Notice that, by using the constant \(c\), we may deduce from this an approximation of \(\zeta(s)\) by a fixed Dirichlet polynomial when \(T\le t\le 2T\), for some parameter \(T\).

2. Size of \(|\zeta(s)|\) and of \(L\)-series

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 \(\displaystyle |\zeta_K(s)|\le 3 \left|\frac{1+s}{1-s}\right|\left(\frac{|d||1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}}\zeta(1+\eta)^n.\)

On the line \(\Re s=1/2\), Lemma 2 of [Lehman, 1970] gives a better result, namely

Theorem (1970)

If \(t\ge 1/5\), we have \(|\zeta(\tfrac12+it)|\le 4 (t/(2\pi))^{1/4}\).

In fact, Lehman states this Theorem for \(t\ge 64/(2\pi)\), but modern means of computations makes it easy to check that it holds as soon as \(t\ge 0.2\). See also equation (56) of [Backlund, 1918] reproduced below.

For Dirichlet \(L\)-series, Theorem 3 of [Rademacher, 1959] gives the corresponding convexity bound.

Theorem (1959)

In the strip \(-\eta\le \sigma\le 1+\eta\), \(0 < \eta\le 1/2\), for all moduli \(q > 1\) and all primitive characters \(\chi\) modulo \(q\), the inequality \(\displaystyle |L(s,\chi)|\le \left(q\frac{|1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}}\zeta(1+\eta)\) holds.

This paper contains other similar convexity bounds.

Corollary to Theorem 3 of [Cheng and Graham, 2004] goes beyond convexity.

Theorem (2001)

For \(0\le t\le e\), we have \(|\zeta(\tfrac12+it)|\le 2.657\). For \(t\ge e\), we have \(|\zeta(\tfrac12+it)|\le 3t^{1/6}\log t\). Section 5 of bibref(“Trudgian*13”) shows that one can replace the constant 3 by 2.38.

This is improved in [Hiary, 2016].

Theorem (2016)

When \(t\ge 3\), we have \(|\zeta(\tfrac12+it)|\le 0.63t^{1/6}\log t\).

Concerning \(L\)-series, the situation is more difficult but [Hiary, 2016] manages, among other and more precise results, to prove the following.

Theorem (2016)

Assume \(\chi\) is a primitive Dirichlet character modulo \(q>1\). Assume further that \(q\) is a sixth power. Then, when \(|t|\ge 200\), we have \(\displaystyle |L(\tfrac12+it,\chi)|\le 9.05d(q) (q|t|)^{1/6}(\log q|t|)^{3/2}\) where \(d(q)\) is the number of divisors of \(q\).

It is often useful to have a representation of the Riemann zeta function or of \(L\)-series inside the critical strip. In the case of \(L\)-series, [Spira, 1969] and [Rumely, 1993] 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 [Odlyzko, 1987]. This expression is due to [Gabcke, 1979]. See also equations (2.4)-(2.5) of [Lehman, 1966], a corrected version of Theorem 2 of [Titchmarsh, 1947].

In general, we have the following estimate taken from equations (53)-(54), (56) and (76) of [Backlund, 1918] (see also [Backlund, 1914]).

Theorem (1918)

• When \(t\ge 50\) and \(\sigma\ge1\), we have \(|\zeta(\sigma+it)|\le \log t-0.048\).

• When \(t\ge 50\) and \(0\le \sigma\le1\), we have \(|\zeta(\sigma+it)|\le\frac{t^2}{t^2-4}\left(\frac{t}{2\pi}\right)^{\frac{1-\sigma}{2}}\log t\).

• When \(t\ge 50\) and \(-1/2\le \sigma\le0\), we have \(|\zeta(\sigma+it)|\le\left(\frac{t}{2\pi}\right)^{\frac{1}{2}-\sigma}\log t\).

On the line \(\Re s=1\), [Trudgian, 2014] establishes the next result.

Theorem (2012)

When \(t\ge 3\), we have \(|\zeta(1+it)|\le\tfrac34 \log t\).

The paper [Patel, 2022] proves the next bound.

Theorem (2022)

When \(t\ge 3\), we have \(|\zeta(1+it)|\le\min\bigl(\frac34 \log t, \frac12\log t + 1.93, \frac15 \log t + 44.02\bigr)\).

Asymptotically better bounds are available since the work of [Ford, 2002].

Theorem (2002)

When \(t\ge 3\) and \(1/2\le \sigma\le 1\), we have \(|\zeta(\sigma+it)|\le 76.2 t^{4.45(1-\sigma)^{3/2} } (\log t)^{2/3}\).

The constants are still too large for this result to be of use in any decent region. See [Kulas, 1994] for an earlier estimate.

3. On the total number of zeroes

The first explicit estimate for the number of zeros of the Riemann \(\zeta\)-function goes back to [Backlund, 1914]. An elegant consequence of the result of Backlund is the following easy estimate taken from Lemma 1 of [Lehman, 1966].

Theorem (1966)

If \(\varphi\) is a continuous function which is positive and monotone decreasing for \(2\pi e\le T_1\le t\le T_2\), then \(\displaystyle \sum_{T_1 < \gamma\le T_2} \varphi(\gamma) =\frac{1}{2\pi}\int_{T_1}^{T_2}\varphi(t)\log\frac{t}{2\pi}dt +O^*\biggl(4\varphi(T_1)\log T_1+2\int_{T_1}^{T_2}\frac{\varphi(t)}{t} dt\biggr)\) where the summation is over all zeros of the Riemann \(\zeta\)-function of imaginary part between \(T_1\) and \(T_2\), with multiplicity.

Theorem 19 of [Rosser, 1941] gives a bound for the total number of zeroes.

Theorem (1941)

For \(T\ge2\), we have \(\displaystyle N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\Bigl(0.137\log T+0.443\log\log T+1.588 \Bigr)\) where the summation is over all zeros of the Riemann \(\zeta\)-function of imaginary part between 0 and \(T\), with multiplicity.

It is noted in Lemma 1 of [Ramaré and Saouter, 2003] that the \(O\)-term can be replaced by the simpler \(O^*(0.67\log\frac{T}{2\pi})\) when \(T\ge 10^3\).

This is improved in Corollary 1 of [Trudgian, 2014] into

Theorem (2014)

For \(T\ge e\), we have \(\displaystyle N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\bigl(0.112\log T+0.278\log\log T+2.510+\frac{1}{5T} \bigr)\) where the summation is over all zeros of the Riemann \(\zeta\)-function of imaginary part between 0 and \(T\), with multiplicity.

Corollary 1.4 of the main theorem of [Hasanalizade et al., 2022] reads

Theorem (2022)

For \(T\ge e\), we have \(\displaystyle N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}}1=\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8}+O^*\bigl(0.1038\log T+0.2573\log\log T+9.3675\bigr)\) where the summation is over all zeros of the Riemann \(\zeta\)-function of imaginary part between 0 and \(T\), with multiplicity. We may also replace \(0.1038\log T+0.2573\log\log T+9.3675\) by \(0.1095\log T+0.2042\log\log T+3.0305\).

Concerning Dirichlet \(L\)-functions, the paper [Bennett et al., 2021] contains the next result.

Theorem (2021)

Let \(\chi\) be a Dirichlet character of conductor \(q > 1\). For \(T\ge 5/7\) and \(\ell= \log\frac{q(T+2)}{2\pi} > 1.567\), we have \(\displaystyle N(T,\chi)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1=\frac{T}{\pi}\log\frac{qT}{2\pi}-\frac{T}{\pi}+\frac{\chi(-1)}{4}+O^*\bigl(0.22737\ell+2\log(1+\ell)-0.5 \bigr)\) where the summation is over all zeros of the Dirichlet function \(L(\cdot,\chi)\) of imaginary part between \(-T\) and \(T\), with multiplicity.

4. \(L^2\)-averages

In Theorem 1.4 of [Kadiri, 2013], we find the next result.

Theorem (2013)

When \(0.5208 < \sigma < 0.9723\) and \(10^3\le H \le 10^{10}\), we have, for any \(T\ge H\), \(\displaystyle \int_H^T |\zeta(\sigma + t)|^2 dt\le (T-H) \bigl(\zeta(2\sigma) +\mathcal{E}_1(\sigma, H)\bigr)\) where \(\mathcal{E}_1(\sigma, H)\) is a small error term whose precise expression in given the stated paper.

We can find in [] the proof of the following estimate. Though it is unpublished yet, the full proof is available.

Theorem (2019)

Let \(0 < \sigma\le1\) and \(T \ge 3\). Then \(\displaystyle \frac{1}{2\pi}\biggl(\int_{\sigma-i\infty}^{\sigma-iT}+ \int^{\sigma+i\infty}_{\sigma+iT} \biggr) \frac{|\zeta(s)|^2}{|s|^2}ds\le\kappa_{\sigma,T} \begin{cases} \frac{c_{1,\sigma}}{T}+\frac{c^\flat_{1,\sigma}}{T^{2\sigma}} &\text{when }\sigma > 1/2,\\ \frac{\log T}{2T}+\frac{c^\flat_{2,\sigma}}{T} &\text{when }\sigma=1/2,\\ c_{3,\sigma}/T^{2\sigma}&\text{when }\sigma < 1/2. \end{cases}\) where \(\displaystyle c_{1,\sigma}=\zeta(2\sigma)/2, c_{1,\sigma}^\flat=c^2 \frac{3^{2\sigma}}{2\sigma}, c_{2,\sigma}^\flat=3c^2+\frac{1-\log 3}{2},c=9/16\) \(\displaystyle c_{3,\sigma}=\Bigl(\frac{c^2}{2\sigma}+\frac{1/6}{1-2\sigma}\Bigr)\Bigl(1+\frac{1}{\sigma}\Bigr)^{2\sigma},\kappa_{\sigma,T}=\begin{cases}\frac{9/4}{\left(1-\frac{9/2}{T^2}\right)^2}&\text{when }1/2\le \sigma\le 1,\\\frac{(1+\sigma)^2}{\left(1-\frac{(1+\sigma)^2}{\sigma T^2}\right)^2}&\text{when }0 < \sigma < 1/2. \end{cases}\)

5. Bounds on the real line

After some estimates from [Bastien and Rogalski, 2002] Lemma 5.1 of [Ramaré, 2016] shows the following.

Theorem (2013)

When \(\sigma> 1\) and \(t\) is any real number, we have \(|\zeta(\sigma+it)|\le e^{\gamma(\sigma-1) }/(\sigma-1)\).

Here is the Theorem of [Delange, 1987]. See also Lemma 2.3 of [Ford, 2000] for a slightly weaker version.

Theorem (1987)

When \(\sigma> 1\) and \(t\) is any real number, we have \(\displaystyle -\Re\frac{\zeta'}{\zeta}(\sigma+it)\le \frac{1}{\sigma-1}-\frac{1}{2\sigma^2}.\)