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We show in this paper that after proper scalings, the characteristicpolynomial of a random unitary matrix converges almost surely to a randomanalytic function whose zeros, which are on the real line, form a determinantalpoint process with sine kernel. Our scaling is performed at the so-called"microscopic" level, that is we consider the characteristic polynomial atpoints which are of order $1/n$ distant. We prove this in the framework ofvirtual isometries to circumvent the fact that the rescaled characteristicpolynomial does not even have a moment of order one, hence making the classicaltechniques of random matrix theory difficult to apply. The strong convergenceresults in this setup provide us with a new approach to ratios: we are able tosolve open problems about the limiting distribution of ratios of characteristicpolynomials evaluated at points of the form $\exp(2 i \pi \alpha/n)$ andrelated objects (such as the logarithmic derivative). We also explicitlydescribe the dependence relation for the logarithm of the characteristicpolynomial evaluated at several points on the microscopic scale. On the numbertheory side, inspired by the Keating-Snaith philosophy, we conjecture some newlimit theorems for the value distribution of the Riemann zeta function on thecritical line at the stochastic process level.

The Circular Unitary Ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios