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We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2}|\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima$\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. Inparticular, we prove that for $(1+o(1))T$ values of $T \leq t \leq 2T$ we have$\max_{|h| \leq 1/2} \log|\zeta(1/2+it+ih)| \leq \log\log T - (3/4 +o(1))\log\log\log T$, matching for the first time with both the leading andsecond order terms predicted by a conjecture of Fyodorov, Hiary and Keating.  The proofs work by approximating the zeta function in mean square by theproduct of a Dirichlet polynomial over smooth numbers and one over roughnumbers. They then apply ideas and results from corresponding random modelproblems to compute averages of this product, under size restrictions on thesmooth part that hold for most $T \leq t \leq 2T$ (but reduce the size of theaverages). There are connections with the study of critical multiplicativechaos. Unlike in some previous work, our arguments never shift away from thecritical line by more than a tiny amount $1/\log T$, and they don't requireexplicit calculations of Fourier transforms of Dirichlet polynomials.

On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture