Arbitrary precision computation of the Riemann-Siegel approximation of the Riemann Zeta function off the critical line

Arbitrary precision computation of the Riemann-Siegel approximation of the Riemann Zeta function off the critical line.

In 1859 a German mathematician named Bernhard Riemann formulated what is considered by some mathematicians to be the most important unresolved problem of pure mathematics, and number theory in particular. It stated that all non-trivial zeroes of the Riemann Zeta function have real part equal to ½. The Riemann Zeta function describes the distribution of prime numbers and has several connections to various fields of physics such as eigenvalue asymptotics of random matrices and operators in quantum physics, the chaos in dynamical systems with random trajectories, and incoherent superposition of waves with random phases. Since the conjecture was originally proposed, several mathematicians have worked towards developing efficient methods in evaluating the function. The most commonly use method is known as the Riemann-Siegel formula. Though our research we hope to implement the Riemann-Siegel formula off the critical line to analyze the Zeta functions properties away from the critical line in order to develop a proof for Riemann’s hypothesis. Furthermore, we hope to use the error function of the Riemann-Siegel formula to determine the required precision by comparing determined error estimates to the actual, known error values.