In this PhD thesis we present an efficient algorithm called RS-PEAK which can be used to find extremely large values of the Riemann zeta function on the critical line. Locating peak values of the zeta function is a promising method for getting a better understanding of the distribution of prime numbers. We investigated multidimensional approximation problems and created an efficient algorithm called MAFRA to solve n-dimensional Diophantine approximation problems. MAFRA was used to generate candidates where large Z(t) values were expected. Using MAFRA and relying on the special behaviour of Z(t) we created the RS-PEAK algorithm using which we were able to locate a high number of large values of the Riemann zeta function on the critical line. The largest Z(t) value found by the RS-PEAK algorithm is Z(310678833629083965667540576593682.05) = 16874.202. To the best of our knowledge, at the time of writing this is the largest Z(t) ever calculated. The value was verified by the ATLAS super computing cluster operating at Eotvos Lorand University, Hungary. In this PhD thesis many new computational results regarding the Z(t) function were published.