Bounds of |ζ(s)|$|\zeta(s)|$ and related questions

Collecting references: [Trudgian, 2011 †Trudgian, T. 2011
Improvements to Turing's method
Math. Comp., 80(276), 2259--2279.
Explicit zero density theorems for Dedekind zeta functions
J. Number Theory, 132(4), 748--775. http://dx.doi.org/10.1016/j.jnt.2011.09.002.
],
1. Size of |ζ(s)|$|\zeta(s)|$ and of L$L$-series
On the Phragmén-Lindelöf theorem and some applications
Math. Z., 72, 192--204.
] gives the convexity bound. See also section 4.1 of [Trudgian, 2014 †Trudgian, Timothy S. 2014
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 ησ1+η$-\eta\le \sigma\le 1+\eta$, 0<η1/2$0 < \eta\le 1/2$, the Dedeking zeta function ζK(s)$\zeta_K(s)$ belonging to the algebraic number field K$K$ of degree n$n$ and discriminant d$d$ satisfies the inequality
|ζK(s)|31+s1s(|d||1+s|2π)1+ησ2ζ(1+η)n.
On the line Rs=1/2$\Re s=1/2$, 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 t1/5$t\ge 1/5$, we have |ζ(12+it)|4(t/(2π))1/4$|\zeta(\tfrac12+it)|\le 4 (t/(2\pi))^{1/4}$.
In fact, Lehman states this Theorem for t64/(2π)$t\ge 64/(2\pi)$, but modern means of computations makes it easy to check that it holds as soon as t0.2$t\ge 0.2$. 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$L$-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 ησ1+η$-\eta\le \sigma\le 1+\eta$, 0<η1/2$0 < \eta\le 1/2$, for all moduli q>1$q > 1$ and all primitive characters χ$\chi$ modulo q$q$, the inequality
|L(s,χ)|(q|1+s|2π)1+ησ2ζ(1+η)
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 0te$0\le t\le e$, we have |ζ(12+it)|2.657$|\zeta(\tfrac12+it)|\le 2.657$. For te$t\ge e$, we have |ζ(12+it)|3t1/6logt$|\zeta(\tfrac12+it)|\le 3t^{1/6}\log t$. Section 5 of [Trudgian, 2014 †Trudgian, Timothy S. 2014
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.
It is often useful to have a representation of the Riemann zeta function or of L$L$-series inside the critical strip. In the case of L$L$-series, [Spira, 1969 †Spira, R. 1969
Calculation of Dirichlet {L$L$}-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.
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.
Sur les zéros de la fonction ζ(s)$\zeta(s)$ de Riemann
C. R. Acad. Sci., 158, 1979--1981.
]).
Theorem (1918)
• When t50$t\ge 50$ and σ1$\sigma\ge1$, we have |ζ(σ+it)|logt0.048$|\zeta(\sigma+it)|\le \log t-0.048$.
• When t50$t\ge 50$ and 0σ1$0\le \sigma\le1$, we have |ζ(σ+it)|t2t24(t2π)1σ2logt$|\zeta(\sigma+it)|\le \frac{t^2}{t^2-4}\left(\frac{t}{2\pi}\right)^{\frac{1-\sigma}{2}}\log t$.
• When t50$t\ge 50$ and 1/2σ0$-1/2\le \sigma\le0$, we have |ζ(σ+it)|(t2π)12σlogt$|\zeta(\sigma+it)|\le \left(\frac{t}{2\pi}\right)^{\frac{1}{2}-\sigma}\log t$.
On the line Rs=1$\Re s=1$, one can rely on [Trudgian, 2012 †Trudgian, T. 2012
A new upper bound for |ζ(1+it)|$|\zeta(1+it)|$
http://arxiv.org/abs/1210.6743.
].
Theorem (2012)
When t3$t\ge 3$, we have |ζ(1+it)|34logt$|\zeta(1+it)|\le\tfrac34 \log t$.
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 t3$t\ge 3$ and 1/2σ1$1/2\le \sigma\le 1$, we have |ζ(σ+it)|76.2t4.45(1σ)3/2(logt)2/3$|\zeta(\sigma+it)|\le 76.2 t^{4.45(1-\sigma)^{3/2} } (\log t)^{2/3}$.
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.
2. On the total number of zeroes

3. L2${}^2$-averages

4. Bounds on the real line
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
], Lemma 5.1 of [Ramaré, 2014 †Ramaré, O. 2014
An explicit density estimate for Dirichlet L$L$-series
To appear in Math. Comp., 35pp.
] shows the following.
Theorem (2013)
When σ>1$\sigma> 1$ and t$t$ is any real number, we have |ζ(σ+it)|eγ(σ1)/(σ1)$|\zeta(\sigma+it)|\le e^{\gamma(\sigma-1) }/(\sigma-1)$.
Here is 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.
]. I have read this estimate in a much older paper, but I am quite unable to remember where!
Theorem (2013)
When σ>1$\sigma> 1$ and t$t$ is any real number, we have ζζ(σ+it)1/(σ1)$\left|\frac{\zeta'}{\zeta}(\sigma+it)\right|\le 1/(\sigma-1)$.

Last updated on September 3rd, 2013, by Olivier Ramaré