geometrical_summation_method

A geometrical summation method for the Riemann zêta function

Codes used for the ArXiv preprint:

https://arxiv.org/pdf/1903.10853.pdf

These codes are written in Python3, and make use of the following libraries:

• numpy 1.15.3
• sympy 1.3
• mpmath 1.1.0
• matplotlib 3.0.3

Overview of the code

radial_convergence_plot.py

This code was used to create some plots described in the article:

For instance, this plot shows in the complex plane (green), the original deverging series (red), and the converging series (blue). In the preprint, this last series is shown to converge to the actual value of zêta.

Uz_calculator.py

This code is meant to compute the approximated expression of Uz for first known non-trivial zeros of the Riemann zêta function. The table of these values is in the preprint, here are the first values: 8, 14, 18, 24, 28, 32, 38, 40, 46...

domination_function.py

This code was used to generate an upper and lower bounds for the function described in the preprint. The expression of these bounds can be written as:

Where q and Q are relative integers. Here is a plot of such a bounding.