Digits of pi: limits to the seeming randomness

TL;DR

This paper investigates the apparent randomness of digits in pi, e, and sqrt(2), revealing that their digit sequences exhibit limited randomness and deviate from expected statistical behaviors over time.

Contribution

It compares the digit behaviors of pi, e, and sqrt(2) with those of powers of 2 in base 3, uncovering limits to their apparent randomness and challenging assumptions about their statistical independence.

Findings

Digits of 2^n in base 3 pass randomness tests inspired by the CLT and LIL.

Digits of pi, e, and sqrt(2) in decimal show convergence to zero, indicating limited randomness.

The apparent randomness of these irrational numbers' digits may be only temporary or limited.

Abstract

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar conjecture is explored - the seemingly almost random behaviour of digits in the base 3 representations of powers $2^n$. This conjecture seems to confirm well - it passes even the tests inspired by the Central Limit Theorem and the Law of the Iterated Logarithm. After this, a similar testing of the sequences of digits in the decimal representations of the numbers $\pi$, $e$ and $\sqrt{2}$ was performed. The result looks surprising: unlike the digits in the base 3 representations of $2^n$, instead of oscillations with amplitudes required by the Law of the Iterated Logarithm, convergence to zero is observed. If, for such "analytically" defined irrational numbers, the observed behaviour remains intact ad infinitum, then the seeming randomness of their digits is only a limited one.