Bounds for $|\zeta(s)|$, $|L(s,\chi)|$ and related questions
Collecting references: , ,
1. Size of $|\zeta(s)|$ and of $L$-series
Theorem 4 of gives the convexity bound. See also section 4.1 of .
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 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 reproduced below. For Dirichlet $L$-series, Theorem 3 of 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 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 shows that one can replace the constant 3 by 2.38.
This is improved in .
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 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, and 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 . This expression is due to . See also equations (2.4)-(2.5) of , a corrected version of Theorem 2 of .
In general, we have the following estimate taken from equations (53)-(54), (56) and (76) of (see also ).
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 $\Re s=1$, one can rely on .
Theorem (2012)
When $t\ge 3$, we have $|\zeta(1+it)|\le\tfrac34 \log t$.
Asymptotically better bounds are available since the huge work of .
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 for an earlier estimate.
2. On the total number of zeroes
The first explicit estimate for the number of zeros of the Riemann $\zeta$-function goes back to . An elegant consequence of the result of Backlund is the following easy estimate taken from Lemma 1 of .
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 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 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 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 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 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.

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

4. Bounds on the real line
After some estimates from , Lemma 5.1 of 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 . See also Lemma 2.3 of for a slightly weaker version.
Theorem (1987)
When $\sigma> 1$ and $t$ is any real number, we have $$ -\Re\frac{\zeta'}{\zeta}(\sigma+it)\le \frac{1}{\sigma-1}-\frac{1}{2\sigma^2}. $$
Last updated on April 29th, 2022, by Olivier Ramaré