The Montgomery-Dyson conjecture roughly states that the statistical properties of the zeros of the Riemann zeta function on the critical line (after rescaling) are similar to those of the eigenvalues of large random unitary matrices (after rescaling). In the past two decades, inspired by this conjecture, and in connection with number theory problems, many new ideas have been emerged in random matrix theory, new results proven and new spectacular conjectures stated (e.g. the moments conjecture by Keating and Snaith). In this talk I shall introduce a new object that we have introduced in 2017 which helped us solve the ratio conjecture in random matrix theory: it is a random holomorphic function whose zero set is a sine kernel point process. In a recent remarkable work, Valko and Virag have called this object the stochastic zeta function and have shown several chacaterizations and extensions of this object. Yet other possible extensions have been recently proved and studied by Lambert and Paquette. I shall focus here on how the idea of this object emerged and how it helps to give a natural solution to the ratio conjecture. I will also discuss an equivalent probabilistic formulation of the GUE conjecture (this is the Montgomery-Dyson conjecture for all correlation levels) involving the stochastic zeta function. In any case, we shall see that the stochastic zeta function (and its generalisations proposed by others) appears as the scaling limit of random analytic functions which have some spectral interpretation. If time allows, I will briefly discuss ongoing work with J. Najnudel where we are able to provide a very general framework/theorem which contains all known convergence results to these generalised stochastic zeta functions.