Extreme values of the Riemann zeta function and its argument

Abstract

We combine our version of the resonance method with certain convolution formulas for $\zeta (s)$ and $\log \zeta (s)$. This leads to a new $\Omega$ result for $|\zeta (1/2+it)|$: The maximum of $|\zeta (1/2+it)|$ on the interval $1\le t\le T$ is at least $\exp \left((1+o(1))\sqrt{\log T\log \log \log T/\log \log T}\right)$. We also obtain conditional results for $S(t):=1/\pi$ times the argument of $\zeta (1/2+it)$ and $S_1(t):={\int }_0^{t}S(\tau )d\tau$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c\sqrt{\log T\log \log \log T/\log \log T}$ and the maximum of $S_1(t)$ is at least $c_1\sqrt{\log T\log \log \log T/(\log \log T{)}^3}$ on the interval $T^{\beta }\le t\le T$ whenever $0\le \beta <1$.

This is the final version of the paper which has been accepted for publication in Mathematische Annalen

References in corpus (1)

• Large GCD sums and extreme values of the Riemann zeta function

Cited by in corpus (8)

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• Maxima of log-correlated fields: some recent developments
• Extreme values of derivatives of zeta and $L$-functions
• A dichotomy for extreme values of zeta and Dirichlet L-functions
• Extreme values for iterated integrals of the logarithm of the Riemann zeta-function
• Large Positive and Negative Values of Hardy's $Z$-Function
• Extreme central values of quadratic Dirichlet $L$-functions with prime conductors