Small values of the Riemann zeta-function near the critical line

Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line. For every fixed $0<k<1\mathrm{/}2$, we establish the conjecturally sharp bound $$\frac{1}{T}{\int }_T^{2T}\mid \zeta (\frac{1}{2}+\alpha +it){\mid }^{-2k}dt\ll {\alpha }^{-k^2},$$ for $\alpha =o(1)$ and $\alpha \gg 1\mathrm{/}\log T$. We also obtain almost sharp upper bounds slightly to the left of $1\mathrm{/}\log T$. This improves the upper bounds of the first two authors for shifts as small as $1\mathrm{/}\log T$, confirming a conjecture of Gonek in this range. A separate bound for the moment at $k=1\mathrm{/}2$ yields an improved bound for cancellation in the Möbius function. Namely, we show that $$\sum _{n\le x}\mu (n)\ll \sqrt{x}\exp \left(C\sqrt{\frac{\log x}{\log \log x}}\right)$$ for some $C>0$, improving on previous results of Soundararajan, Balazard and de Roton, and of the first two authors. For $t$ chosen uniformly from $[T,2T]$, we also prove that $$\underset{\mathrm{\mid }h\mathrm{\mid }\le 1}{\min }\mid \zeta (\frac{1}{2}+\alpha +i(t+h))\mid ={\alpha }^{1+o(1)}$$ in probability, provided that $\alpha \ll (\log T{)}^{-c}$ for some fixed $c>0$, and $\alpha \log T\mathrm{/}\log \log T\to \mathrm{\infty }$. This agrees with the leading order predicted by a minimum analogue of the Fyodorov–Hiary–Keating Conjecture. Further results include bounds for minima on intervals of logarithmically growing length, for negative moments in almost all short intervals, and an extension of the range of sharp negative-moment bounds for $k>1\mathrm{/}2$. A main ingredient in the proof of some of these results is a Gaussian large-deviation upper bound for $\log \mathrm{\mid }\zeta (1\mathrm{/}2+\alpha +it){\mathrm{\mid }}^{-1}$, obtained by building on work of Arguin, Bailey, and Roberts.