A formula for the $n^{\rm th}$ decimal digit or binary of $\pi$ and powers of $\pi$

TL;DR

This paper presents an explicit formula derived from asymptotic analysis that allows computation of the nth decimal or binary digit of π and its powers, providing a new method for digit extraction.

Contribution

It introduces a novel explicit formula based on asymptotic formulas for Euler and Bernoulli numbers to compute specific digits of π and its powers.

Findings

Explicit formula for the nth digit of π in decimal and binary

Method to compute the nth digit of powers of π

Potential for efficient digit extraction algorithms

Abstract

By using an asymptotic formula known for the numbers of Euler and Bernoulli it is possible to obtain an explicit expression of the nth digit of $\pi$ in decimal or in binary, it also makes it possible to obtain the $n^{\rm th}$ digit of powers of $\pi^n$.