Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance:• its value in 1 (Keating-Snaith theorem),• the truncation of its Fourier series up to any fraction of its degree,• the computation of the relative volume of the Birkhoff polytope,• its products and ratios taken in different points,• the product of its iterated derivatives in different points,• functionals in relation with sums of divisor functions in F q [X].• its mid-secular coefficients,• the "moments of moments", etc.We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional. Y. BARHOUMI-ANDRÉANI 2. Notations and prerequisites 2.1. General notations and conventions 2.2. Reminders on symmetric functions 2.3. Scalar products and unitary integrals 2.4. Classical tricks 3. Theory 3.1. The ubiquitous Schur function 3.2. The Schur-CUE connection 3.3. A plethystic-RKHS perspective on duality 3.4. CFKRS with RKHS 3.5. Duality in H N 4. Applications 4.1. The Keating-Snaith theorem 4.2. Autocorrelations of the characteristic polynomial in the microscopic setting 4.3. Ratios of the characteristic polynomial in the microscopic setting 4.4. The mid-secular coefficients 4.5. Back to the autocorrelations : the randomisation paradigm 4.6. The truncated characteristic polynomial 4.7. The Birkhoff polytope 4.8. Iterated derivatives of the characteristic polynomial 4.9. Sums of divisor functions in F q [X] 4.10. The moments of moments 5. Ultimate remarks 5.1. The Hankel form of the limit 5.2. Expansions 5.3. Other approaches to s λ [A] 5.4. Summary of the encountered functionals and limits 5.5. Partial conclusion and future work Appendix A. Probabilistic representations A.1. A symmetric function extension of Gegenbauer polynomials A.2. An integral representation of h n [A] A.3. Probabilistic representations of h (κ) c,∞ A.4. Integrability of h (κ) c,∞ A.5. Integrability in c A.6. The supersymmetric case Appendix B. Truncated ζ function in the microscopic scaling B.1. Reminders B.2. A phase transition Acknowledgements References

We show that for any linear combination of characteristic polynomials of independent random unitary matrices with the same determinant, the expected proportion of zeros lying on the unit circle tends to 1 as the dimension of the matrices tends to infinity. This result is the random matrix analog of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of L-functions, and thus is consistent with the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of L-functions on the critical line.

We study arithmetic progressions {a, a+b, a+2b, . . . , a+(ℓ−1)b}, with ℓ ≥ 3, in random subsets of the initial segment of natural numbers [n] := {1, 2, . . . , n}. Given p ∈ [0, 1] we denote by [n]p the random subset of [n] which includes every number with probability p, independently of one another. The focus lies on sparse random subsets, i.e. when p = p(n) = o(1) as n → +∞.Let X ℓ denote the number of distinct arithmetic progressions of length ℓ which are contained in [n]p. We determine the limiting distribution for X ℓ not only for fixed ℓ ≥ 3 but also when ℓ = ℓ(n) → +∞. The main result concerns the joint distribution of the pair (X ℓ , X ℓ ′ ), ℓ > ℓ ′ , for which we prove a bivariate central limit theorem for a wide range of p. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or nontrivial is characterised by the asymptotic behaviour (as n → +∞) of the threshold function ψ ℓ = ψ ℓ (n) := np ℓ−1 ℓ. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

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