Detecting Subtle Departures from Randomness

 I discuss a new test of randomness for pseudo random number generators (PRNG), to detect subtle patterns in binary sequences. The test shows that congruential PRNGs, even the best ones, have flaws that can be exacerbated by the choice of the seed. This includes the Mersenne twister used in many programming languages including Python. I also show that the digits of some numbers, conjectured to be perfectly random, fail this new test, despite the fact that they pass all the standard tests. I propose a methodology to avoid these flaws, implemented in Python. The test is particularly useful when high quality randomness is needed. This includes cryptographic and military-grade security applications, as well as synthetic data generation and simulation-intensive Markov chain Monte Carlo methods. The origin of this test is in number theory and connected to the Riemann Hypothesis. In particular, it is based on Rademacher stochastic processes. These random multiplicative functions are a number-theoretic version of Bernoulli trials. My article features state-of-the-art research on this topic, as well as an original, simple, integer-based formula to compute square roots to generate random digits. It is offered with a Python implementation that handles integers with millions of digits.