This thesis is divided into three parts. In the first part we study the uniformity in the shifts in the asymptotic formulae for the second moment of the Riemann zeta-function and the first moments of the Hecke and the quadratic Dirichlet L-functions. In the second part we investigate the period function of the Eisenstein series. We use our results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moment of the Riemann zeta function. Moreover, we study a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. In the third part, we find optimal Dirichlet polynomials for the Nyman- Beurling criterion for the Riemann- hypothesis, conditionally on some separa- tion condition on the zeros of ((8) and on the Riemann hypothesis.