On the sum of digits of $1/M$ in $\mathbb{F}_q[x]$

TL;DR

This paper investigates the digit sum behavior of 1/M in polynomial rings over finite fields, revealing no bias and contrasting with known phenomena over integers related to class numbers.

Contribution

It demonstrates that, unlike the integer case, the polynomial case over finite fields exhibits no bias in digit sums, using an elementary argument.

Findings

No bias in digit sums of 1/M over finite fields.

Contrast with integer case where bias relates to class numbers.

Elementary proof technique used.

Abstract

For certain primes $p$, the average digit in the expansion of $1/p$ was found to have a deviation from random behaviour related to the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ (Girstmair 1994). In this short note, we observe that for the corresponding problem when we replace the integers by polynomials over a finite field, there is never any bias. The argument is elementary.