An incremental O(√x) algorithm estimates π(x) via generalized triangular number partitions and a numerically fitted correction term that matches known values up to 10^19.
Heath-Brown, The number of primes in a short interval
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An Efficient Algorithm for Estimating Prime Counts
An incremental O(√x) algorithm estimates π(x) via generalized triangular number partitions and a numerically fitted correction term that matches known values up to 10^19.