**Keywords:**Riemann zeta-function | Riemann hypothesis | argument | critical strip | Beurling-Selberg extremal problem | extremal functions | Gaussian subordination | exponential type | extrem values | pair correlation of the zeros.

In this Ph.D. thesis, we establish new bounds for some objects related to the Riemann zeta-function and L-functions, under the Riemann hypothesis, making use of ne tools from analytic number theory, harmonic analysis, and approximation theory. Firstly, we use extremal bandlimited approximations to show bounds for the high moments of the argument of the Riemann zeta-function and for a family of L-functions. Secondly, we use the resonance method of Soundararajan, in the version of Bondarenko and Seip, to obtain large values for the high moments of the argument function. Finally, we improve some estimates related with the distribution of the zeros of the Riemann zeta-function, using the approach of pair correlation of Montgomery and tools from semidefinite programming.