Large Oscillations of the Argument of the Riemann Zeta-function

Andrés Chirre; Kamalakshya Mahatab

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arXiv · 1904.11051

Large Oscillations of the Argument of the Riemann Zeta-function

Andrés Chirre·Kamalakshya Mahatab

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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.

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Andrés Chirre, Kamalakshya Mahatab. 2019-04-24. Large Oscillations of the Argument of the Riemann Zeta-function. https://arxiv.org/abs/1904.11051

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