Computation of the n-th decimal digit of π with low memory

This paper presents an algorithm that computes directly the n-th decimal digit of π in sub-quadratic time and very low memory. It improves previous results of Simon Plouffe, later refined by Fabrice Bellard. The problem of the n-th digit computation in base 2 had already been successfully treated thanks to the use of appropriate series, but no corresponding formula for the question in base 10 has been found yet. However, our result is a progress. Another result in this paper permits to compute directly the n-th decimal digit of π with intermediate memory size, leading to intermediate time complexity between linear and quadratic.