# Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Thomas Christ[1]; Justas Kalpokas[2]

• [1] Department of Mathematics Würzburg University Emil-Fischer-Str. 40, 97074 Würzburg
• [2] Faculty of Mathematics and Informatics Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania
• Volume: 25, Issue: 2, page 285-305
• ISSN: 1246-7405

top

## Abstract

top
We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments ${I}_{k,l}\left(T\right)={\int }_{0}^{T}{|{\zeta }^{\left(l\right)}\left(\frac{1}{2}+it\right)|}^{2k}dt$, where $l$ is a non-negative integer and $k\ge 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for ${I}_{k,l}\left(T\right)$.

## How to cite

top

Christ, Thomas, and Kalpokas, Justas. "Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line." Journal de Théorie des Nombres de Bordeaux 25.2 (2013): 285-305. <http://eudml.org/doc/275751>.

@article{Christ2013,
abstract = {We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_\{k,l\}(T)=\int _\{0\}^\{T\}|\zeta ^\{(l)\}(\frac\{1\}\{2\}+it)|^\{2k\}dt$, where $l$ is a non-negative integer and $k\ge 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_\{k,l\}(T)$.},
affiliation = {Department of Mathematics Würzburg University Emil-Fischer-Str. 40, 97074 Würzburg; Faculty of Mathematics and Informatics Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania},
author = {Christ, Thomas, Kalpokas, Justas},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Riemann zeta-function; discrete moment; lower bound},
language = {eng},
month = {9},
number = {2},
pages = {285-305},
publisher = {Société Arithmétique de Bordeaux},
title = {Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line},
url = {http://eudml.org/doc/275751},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Christ, Thomas
AU - Kalpokas, Justas
TI - Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/9//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 2
SP - 285
EP - 305
AB - We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k,l}(T)=\int _{0}^{T}|\zeta ^{(l)}(\frac{1}{2}+it)|^{2k}dt$, where $l$ is a non-negative integer and $k\ge 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k,l}(T)$.
LA - eng
KW - Riemann zeta-function; discrete moment; lower bound
UR - http://eudml.org/doc/275751
ER -

## References

top
1. T. Christ, J. Kalpokas, Upper bounds for discrete moments of the derivatives of the riemann zeta-function on the critical line, Lith. Math. Journal52 (2012) 233–248. Zbl1322.11089MR3020940
2. R.D. Dixon, L. Schoenfeld, The size of the Riemann zeta-function at places symmetric with respect to the point $1/2$, Duke Math. J.33 (1966), 291–292. Zbl0154.04601MR190103
3. H.M. Edwards, Riemann’s zeta function, Academic Press, New York 1974. Zbl0315.10035MR466039
4. S.M. Gonek, Mean values of the Riemann zeta-function and its derivatives, Invent. Math.75:1 (1984), 123–141. Zbl0531.10040MR728143
5. J. Gram, Sur les zéros de la fonction $\zeta \left(s\right)$ de Riemann, Acta Math.27 (1903), 289–304. Zbl34.0461.01MR1554986
6. G.H. Hardy, J.E. Littlewood, Contributions to the theory of the Riemann zeta-function, Proc. Royal Soc. (A)113 (1936), 542–569.
7. D. R. Heath-Brown, Fractional Moments of the Riemann Zeta-Function J. London Math. Soc.2-24 (1981) 65–78. Zbl0431.10024MR623671
8. A.E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. (2)27 (1926), 273–300. Zbl53.0313.01MR1575391
9. A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York 1985. Zbl0556.10026MR792089
10. J. Kalpokas, J. Steuding, On the Value-Distribution of the Riemann Zeta-Function on the Critical Line, Moscow Jour. Combinatorcs and Number Theo.1 (2011), 26–42. Zbl1302.11060MR2948324
11. J. Kalpokas, M. Korolev, J. Steuding, Negative values of the Riemann Zeta-Function on the Critical Line, preprint, available at arXiv:1109.2224. Zbl1292.11094
12. M.B. Milinovich, N. Ng, Lower bound for the moments of ${\zeta }^{\prime }\left(\rho \right)$, preprint, available at arXiv:0706.2321v1. Zbl1296.11109
13. M.B. Milinovich, Moments of the Riemann zeta-function at its relative extrema on the critical line, preprint, available at arXiv:1106.1154. MR2861534
14. N. Ng, A discrete mean value of the derivative of the Riemann zeta function, Mathematika54 (2007), 113–155. Zbl1170.11026MR2428211
15. M. Radziwill, The 4.36th moment of the Riemann Zeta-function, Int. Math. Res. Not.18 (2012), 4245–4259. Zbl1290.11120MR2975381
16. K. Ramachandra, Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series , Hardy-Ramanujan J. I 1 (1978), 1–15. Zbl0411.10013MR565298
17. Z. Rudnick, K. Soundararajan, Lower bounds for moments of L-functions, Proc. Natl. Sci. Acad. USA102 (2005), 6837–6838. Zbl1159.11317MR2144738
18. R. Spira, An inequality for the Riemann zeta function, Duke Math. J.32 (1965), 247–250. Zbl0154.04501MR176964
19. K. Soundararajan Moments of the Riemann zeta function, Ann. Math. (2) 170, No. 2, 981-993 (2009) Zbl1251.11058MR2552116
20. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced mathematics 46, Cambridge University Press 1995. Zbl0831.11001MR1342300
21. E.C. Titchmarsh, The Riemann zeta-function, 2nd edition, revised by D.R. Heath-Brown, Oxford University Press 1986. Zbl0601.10026MR882550

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.