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

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

On the total number of zeroes

L${}^2$-averages

Bounds on the real line

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

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Collecting references: [Trudgian, 2011 †Trudgian, T. 2011
Improvements to Turing's method
Math. Comp., 80(276), 2259--2279.], [Kadiri & Ng, 2012 †Kadiri, H., & Ng, N. 2012
Explicit zero density theorems for Dedekind zeta functions
J. Number Theory, 132(4), 748--775.],
In [Arias de Reyna, 2011 †Arias de Reyna, J. 2011
High precision computation of Riemann's zeta function by the Riemann-Siegel formula, I
Math. Comp., 80(274), 995--1009.] extends the Phd memoir [Gabcke, 1979 †Gabcke, W. 1979
Neue Herleitung und explizite Restabschaetzung der Riemann-Siegel-Formel
Ph.D. thesis, Mathematisch-Naturwissenschaftliche Fakultät der Georg-August-Universität zu Göttingen.] and provides an explicit Riemann-Siegel formula for $\zeta(s)$. Theorem 1.2 of [Kadiri, 2013 †Kadiri, H. 2013
A zero density result for the Riemann zeta function
Acta Arith., 160(2), 185--200.] 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 $$ \zeta(s) =\sum_{n < c t} \frac{1}{n^s} + \mathcal{O}^* \biggl( \biggl(c+\tfrac12+\frac{3\sqrt{1+1/t_0^2}}{2\pi} \biggl(\frac{\pi}{12c}+1+\frac{1}{2\pi c-1}\biggr) \biggr) (ct)^{-\sigma}\biggr). $$

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 $T$.
Theorem 4 of [Rademacher, 1959 †Rademacher, H. 1959
On the Phragmén-Lindelöf theorem and some applications
Math. Z., 72, 192--204.] gives the convexity bound. See also section 4.1 of [Trudgian, 2014b †Trudgian, Timothy S. 2014b
An improved upper bound for the argument of the Riemann zeta-function on the critical line II
J. Number Theory, 134, 280--292.].
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 $$ |\zeta_K(s)|\le 3 \left|\frac{1+s}{1-s}\right| \left(\frac{|d||1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}} \zeta(1+\eta)^n. $$

On the line $\Re s=1/2$, Lemma 2 of [Lehman, 1970 †Lehman, R.S. 1970
On the distribution of zeros of the Riemann zeta-function
Proc. London Math. Soc. (3), 20, 303--320.] 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 †Backlund, R. J. 1918
Über die Nullstellen der {\it Riemannschen Zetafunktion.}
Acta Math., 41, 345--375.] reproduced below. For Dirichlet $L$-series, Theorem 3 of [Rademacher, 1959 †Rademacher, H. 1959
On the Phragmén-Lindelöf theorem and some applications
Math. Z., 72, 192--204.] 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 $$ |L(s,\chi)|\le \left(q\frac{|1+s|}{2\pi}\right)^{\frac{1+\eta-\sigma}{2}} \zeta(1+\eta) $$ holds.

This paper contains other similar convexity bounds. Corollary to Theorem 3 of [Cheng & Graham, 2004 †Cheng, Y., & Graham, S.W. 2004
Explicit estimates for the Riemann zeta function
Rocky Mountain J. Math., 34(4), 1261--1280.] 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 [Trudgian, 2014b †Trudgian, Timothy S. 2014b
An improved upper bound for the argument of the Riemann zeta-function on the critical line II
J. Number Theory, 134, 280--292.] shows that one can replace the constant 3 by 2.38.

This is improved in [Hiary, 2016b †Hiary, Ghaith A. 2016b
An explicit van der Corput estimate for {$\zeta(1/2+it)$}
Indag. Math. (N.S.), 27(2), 524--533.].
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, 2016a †Hiary, Ghaith A. 2016a
An explicit hybrid estimate for {$L(1/2+it,\chi)$}
Acta Arith., 176(3), 211--239.] 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 $$|L(\tfrac12+it,\chi)|\le 9.05d(q) (q|t|)^{1/6}(\log q|t|)^{3/2}$$ 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 †Spira, R. 1969
Calculation of Dirichlet {$L$}-functions
Math. Comp., 23, 489--497.] and [Rumely, 1993 †Rumely, R. 1993
Numerical Computations Concerning the ERH
Math. Comp., 61, 415--440.] 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 †Odlyzko, A.M. 1987
On the distribution of spacings between zeros of the zeta function
Math. Comp., 48(177), 273--308.]. This expression is due to [Gabcke, 1979 †Gabcke, W. 1979
Neue Herleitung und explizite Restabschaetzung der Riemann-Siegel-Formel
Ph.D. thesis, Mathematisch-Naturwissenschaftliche Fakultät der Georg-August-Universität zu Göttingen.]. See also equations (2.4)-(2.5) of [Lehman, 1966b †Lehman, R.S. 1966b
Separation of zeros of the Riemann zeta-function
Math. Comp., 20, 523--541.], a corrected version of Theorem 2 of [Titchmarsh, 1947 †Titchmarsh, E.C. 1947
On the zeros of the Riemann zeta function
Quart. J. Math., Oxford Ser., 18, 4--16.].
In general, we have the following estimate taken from equations (53)-(54), (56) and (76) of [Backlund, 1918 †Backlund, R. J. 1918
Über die Nullstellen der {\it Riemannschen Zetafunktion.}
Acta Math., 41, 345--375.] (see also [Backlund, 1914 †Backlund, R.J. 1914
Sur les zéros de la fonction $\zeta(s)$ de Riemann
C. R. Acad. Sci., 158, 1979--1981.]).
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, 2014a †Trudgian, Timothy. 2014a
A new upper bound for {$|\zeta(1+it)|$}
Bull. Aust. Math. Soc., 89(2), 259--264.] establishes the next result.
Theorem (2012)

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

The paper [Patel, 2022 †Patel, Dhir. 2022
An explicit upper bound for {$|\zeta (1+it) |$}
Indag. Math. (N.S.), 33(5), 1012--1032.] proves the next bound.
Theorem (2022)

When $t\ge 3$, we have $ |\zeta(1+it)|\le\min\bigl(\tfrac34 \log t, \tfrac12\log t + 1.93, \tfrac15 \log t + 44.02\bigr). $

Asymptotically better bounds are available since the huge work of [Ford, 2002 †Ford, K. 2002
Vinogradov's integral and bounds for the Riemannn zeta function
Proc. London Math. Soc., 85, 565--633.].
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 †Kulas, M. 1994
Some effective estimation in the theory of the Hurwitz-zeta function
Funct. Approx. Comment. Math., 23, 123--134 (1995).] for an earlier estimate.
The first explicit estimate for the number of zeros of the Riemann $\zeta$-function goes back to [Backlund, 1914 †Backlund, R.J. 1914
Sur les zéros de la fonction $\zeta(s)$ de Riemann
C. R. Acad. Sci., 158, 1979--1981.]. An elegant consequence of the result of Backlund is the following easy estimate taken from Lemma 1 of [Lehman, 1966a †Lehman, R. Sherman. 1966a
On the difference {$\pi (x)-{\rm li}(x)$}
Acta Arith., 11, 397--410.].
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 $$ \sum_{T_1 < \gamma\le T_2} \varphi(\gamma) =\frac{1}{2\pi}\int_{T_1}^{T_2}\varphi(t)\log\frac{t}{2\pi}dt +O^*\biggl(4\varphi(T_1)\log T_1+2\int_{T_1}^{T_2}\frac{\varphi(t)}{t} dt\biggr) $$ 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 †Rosser, J.B. 1941
Explicit bounds for some functions of prime numbers
Amer. J. Math., 63, 211--232.] gives a bound for the total number of zeroes.
Theorem (1941)

For $T\ge2$, we have $$ N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\Bigl(0.137\log T+0.443\log\log T+1.588 \Bigr) $$ 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 & Saouter, 2003 †Ramaré, O., & Saouter, Y. 2003
Short effective intervals containing primes
J. Number Theory, 98, 10--33.] 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, 2014b †Trudgian, Timothy S. 2014b
An improved upper bound for the argument of the Riemann zeta-function on the critical line II
J. Number Theory, 134, 280--292.] into
Theorem (2014)

For $T\ge e$, we have $$ N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\bigl(0.112\log T+0.278\log\log T+2.510+\frac{1}{5T} \bigr) $$ 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 †Hasanalizade, Elchin, Shen, Quanli, & Wong, Peng-Jie. 2022
Counting zeros of the Riemann zeta function
J. Number Theory, 235, 219--241.] reads
Theorem (2022)

For $T\ge e$, we have $$ N(T)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{7}{8} +O^*\bigl(0.1038\log T+0.2573\log\log T+9.3675 \bigr) $$ 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 †Bennett, Michael A., Martin, Greg, O'Bryant, Kevin, & Rechnitzer, Andrew. 2021
Counting zeros of Dirichlet {$L$}-functions
Math. Comp., 90(329), 1455--1482.] 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 $$ N(T,\chi)=\sum_{\substack{\rho,\\ 0 < \gamma\le T}} 1= \frac{T}{\pi}\log\frac{qT}{2\pi}-\frac{T}{\pi}+\frac{\chi(-1)}{4} +O^*\bigl(0.22737\ell+2\log(1+\ell)-0.5 \bigr) $$ where the summation is over all zeros of the Dirichlet function $L(\cdot,\chi)$ of imaginary part between $-T$ and $T$, with multiplicity.

In Theorem 1.4 of [Kadiri, 2013 †Kadiri, H. 2013
A zero density result for the Riemann zeta function
Acta Arith., 160(2), 185--200.], 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 $T\ge H$, $$ \int_H^T |\zeta(\sigma + t)|^2 dt \le (T-H) \bigl(\zeta(2\sigma) + \mathcal{E}_1(\sigma, H)\bigr) $$ where $\mathcal{E}_1(\sigma, H)$ is a small error term whose precise expression in given the stated paper.

We can find in [Helfgott, 2017 †Helfgott, H.A. 2017
$L^2$ bounds for tails of $\zeta(s)$ on a vertical line
.] 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 $$ \frac{1}{2\pi}\biggl( \int_{\sigma-i\infty}^{\sigma-iT} + \int^{\sigma+i\infty}_{\sigma+iT} \biggr) \frac{|\zeta(s)|^2}{|s|^2}ds\le \kappa_{\sigma,T} \begin{cases} \frac{c_{1,\sigma}}{T}+\frac{c^\flat_{1,\sigma}}{T^{2\sigma}} &\text{when $\sigma > 1/2$,}\\ \frac{\log T}{2T}+\frac{c^\flat_{2,\sigma}}{T} &\text{when $\sigma=1/2$,}\\ c_{3,\sigma}/T^{2\sigma}&\text{when $\sigma < 1/2$.} \end{cases} $$ where $$ c_{1,\sigma}=\zeta(2\sigma)/2, c_{1,\sigma}^\flat=c^2 \frac{3^{2\sigma}}{2\sigma}, c_{2,\sigma}^\flat=3c^2+\frac{1-\log 3}{2}, c=9/16, $$ $$ c_{3,\sigma}= \Bigl(\frac{c^2}{2\sigma}+\frac{1/6}{1-2\sigma}\Bigr) \Bigl(1+\frac{1}{\sigma}\Bigr)^{2\sigma}, \kappa_{\sigma,T}= \begin{cases} \frac{9/4}{\left(1-\frac{9/2}{T^2}\right)^2} &\text{when $1/2\le \sigma\le 1$,}\\ \frac{(1+\sigma)^2}{\left(1-\frac{(1+\sigma)^2}{\sigma T^2}\right)^2} &\text{when $0 < \sigma < 1/2$.} \end{cases} $$

After some estimates from [Bastien & Rogalski, 2002 †Bastien, G., & Rogalski, M. 2002
Convexité, complète monotonie et inégalités sur les fonctions z\^eta et gamma, sur les fonctions des opérateurs de Baskakov et sur des fonctions arithmétiques
Canad. J. Math., 54(5), 916--944.], Lemma 5.1 of [Ramaré, 2016 †Ramaré, O. 2016
An explicit density estimate for Dirichlet $L$-series
Math. Comp., 85(297), 335--356.] 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 †Delange, Hubert. 1987
Une remarque sur la dérivée logarithmique de la fonction z\^eta de Riemann
Colloq. Math., 53(2), 333--335.]. See also Lemma 2.3 of [Ford, 2000 †Ford, K. 2000
Zero-free regions for the Riemann zeta function
Proceedings of the Millenial Conference on Number Theory, Urbana, IL.] for a slightly weaker version.
Theorem (1987)

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

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