Phase Portraits of Bi-dimensional Zeta Values

摘要

In this extended abstract, we present how to compute and visualize phase portraits of bi-dimensional Zeta Values. Such technology is useful to explore bi-dimensional Zeta Values and in long-term quest to discover a 2D-Riemann hypothesis. To reach this goal, we need two preliminary steps: 1. the notion of phase portraits and a general tool to visualize phase portrait based on interactive Jupyter widgets. 2. the ability to compute numerical approximations of bi-dimensional Zeta values, using mpmath, a Python library for arbitrary-precision floating-point arithmetic. To this end, we develop a theory to numerically compute double sums and produce the first algorithm to compute bi-dimensional Zeta Values with complex parameters.