Exact computations of the number of primes

Collecting references: [Deléglise and Rivat, 1998], [Platt, 2011].

Papers [Deléglise and Rivat, 1996] and Silva, 2006 present tables of values of \(\pi(x)\) for powers of \(10\) up to \(10^{18}\) and \(10^{22}\), respectively. The later also computed \(\pi(x)\) for \(x = 2^k\) for \(k \leq 72\). [Platt, 2011] also computed \(\pi(x)\) for \(x = 10^{23}\). In Deléglise et al, 2004, the authors computes the functions \(\pi(x,4,1)\) and \(\pi(x,4,3)\) that counts the number of prime numbers up to \(x\) in the residue classes \(p \equiv 1 \mod 4\) and \(p \equiv 3 \mod 4\), respectively, for \(x = m \times 10^k\) for \(m = 1, 2, \dots, 9\) and \(k = 10, 11, \dots, 20\) (for \(k = 20\), only \(m = 1\) was considered).

Paper [Deléglise and Rivat, 1996] computes the Chebyshev function \(\psi(x)\) for \(x = m \times 10^k\) for \(m = 1, 2, \dots, 9\) and \(k = 6, 7, \dots, 15\) (for \(k = 15\), only \(m = 1\) was considered).