Using Fourier analysis, we study local limit theorems in
weak-convergence problems. Among many applications, we discuss random matrix
theory, some probabilistic models in number theory, the winding number of
complex brownian motion and the classical situation of the central limit
theorem. Another application, is to a conjecture of Ramachandra that the
values of the Riemann zeta function on the critical line is dense in complex
numbers, assuming a weak form of a conjecture of Keating and Snaith.
