Large Oscillations of the Argument of the Riemann Zeta-function

Andrés Chirre; Kamalakshya Mahatab

Large Oscillations of the Argument of the Riemann Zeta-function

Number Theory 2021-06-02 v4

Authors: Andrés Chirre , Kamalakshya Mahatab

Abstract

Let $S(t)$ denote the argument of the Riemann zeta-function, defined as $S(t)=\dfrac{1}{\pi}\,\Im\log\zeta(1/2+it).$ Assuming the Riemann hypothesis, we prove that $S(t)=\Omega_{\pm}\bigg(\dfrac{\log t\log\log\log t}{\log\log t}\bigg).$ This improves the classical omega results of Montgomery and matches with the $\Omega$ -result obtained by Bondarenko and Seip.

Keywords

riemann zeta-function argument large oscillations riemann hypothesis omega results

Cite

@article{arxiv.1904.11051,
  title  = {Large Oscillations of the Argument of the Riemann Zeta-function},
  author = {Andrés Chirre and Kamalakshya Mahatab},
  journal= {arXiv preprint arXiv:1904.11051},
  year   = {2021}
}

Comments

9 pages. We corrected a mistake in the previous version and added Andr\'{e}s Chirre as a coauthor. To appear in Bull. Lond. Math. Soc

Large Oscillations of the Argument of the Riemann Zeta-function

Abstract

Keywords

Cite

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