Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T23:06:15.779Z Has data issue: false hasContentIssue false

A NOTE ON THE LARGE VALUES OF $|\zeta ^{(\ell )}(1+{i}t)|$

Published online by Cambridge University Press:  05 January 2023

ZIKANG DONG
Affiliation:
CNRS LAMA 8050, Laboratoire d’analyse et de mathématiques appliquées, Université Paris-Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France e-mail: zikangdong@gmail.com
BIN WEI*
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin 300072, PR China
*

Abstract

We investigate the large values of the derivatives of the Riemann zeta function $\zeta (s)$ on the 1-line. We give a larger lower bound for $\max _{t\in [T,2T]}|\zeta ^{(\ell )}(1+{i} t)|$, which improves the previous result established by Yang [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510].

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The first author is supported by the China Scholarship Council (CSC) for his study in France. The second author is supported by Natural Science Foundation of Tianjin City (Grant No. 19JCQNJC14200).

References

Bondarenko, A. and Seip, K., ‘ Note on the resonance method for the Riemann zeta function ’, in: 50 Years with Hardy Spaces, Operator Theory: Advances and Applications, 261 (eds. Baranov, A., Kisliakov, S. and Nikolski, N.) (Birkhäuser/Springer, Cham, 2018), 121139.10.1007/978-3-319-59078-3_6CrossRefGoogle Scholar
Dong, Z. and Wei, B., ‘On large values of $\mid \zeta \left(\sigma +\mathrm{it}\right)\mid$ ’, Preprint, 2022, arXiv:2110.04278.Google Scholar
Granville, A. and Soundararajan, K., ‘Extreme values of $\mid \zeta \left(1+\mathrm{it}\right)\mid$ ’, in: The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra, Ramanujan Mathematical Society Lecture Notes Series, 2 (eds. Balsubramanian, R. and Srinivas, K.) (International Press, Mysore, 2006), 6580.Google Scholar
Littlewood, J. E., ‘On the Riemann zeta-function’, Proc. Lond. Math. Soc. (2) 24 (1925), 175201.Google Scholar
Soundararajan, K., ‘Extreme values of zeta and $L$ -functions’, Math. Ann. 342 (2008), 467486.CrossRefGoogle Scholar
Titchmarsh, E. C., ‘On an inequality satisfied by the zeta-function of Riemann’, Proc. Lond. Math. Soc. 28 (1928), 7080.10.1112/plms/s2-28.1.70CrossRefGoogle Scholar
Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, 2nd edn (Oxford University Press, New York, 1986).Google Scholar
Yang, D., ‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486510.10.1112/mtk.12130CrossRefGoogle ScholarPubMed