This thesis shows that Gram's Law and the Rosser Rule (methods for locating zeroes of the Riemann zeta-function) fail in a positive proportion of cases. A weaker version of Gram's Law is shown to be true in a positive proportion of cases. Also included are theorems on Turing's Method and its extensions to Dirichlet L-functions and Dedekind zeta-functions.

In 1952 Alan Turing published his paper "The Chemical Basis of Morphogenesis", which described a model for how naturally occurring patterns, such as the stripes of a zebra and the spots of a leopard, can arise from a spatially homogeneous steady state through diffusion. Turing suggested that the concentration of the substances producing the patterns is determined by the reaction kinetics, how the substances interact, and diffusion.  In this project Turing's model with linear reactions kinetics was studied. The model was first solved using two different numerical methods; the finite difference method (FDM) and the finite element method (FEM) with different boundary conditions. A parameter study was then conducted, investigating the effect on the patterns of changing the parameters of the model. Lastly the controllability of the model and the least energy control was considered. The simulations were found to produce patterns provided the right parameters, as expected. From the investigation of the parameters it could be concluded that the size/tightness of the pattern and similarity of the substance concentration distributions depended on the choice of parameters. As for the controllability, a desired final state could be produced thorough simulations using control of the boundary and the energy cost of producing the pattern increased when decreasing the number of controls.