Exact computations of the number of primes
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Collecting references:
:cite:`Deleglise-Rivat98`,
:cite:`Platt11`.


Papers  :cite:`Deleglise-Rivat96-1`
and `Silva, 2006 <Biblio.html#silva-06>`_ present tables of values of :math:`\pi(x)` for powers of :math:`10` 
up to :math:`10^{18}` and :math:`10^{22}`, respectively. The later also computed :math:`\pi(x)` for :math:`x = 2^k`
for :math:`k \leq 72`. :cite:`Platt11` also computed  :math:`\pi(x)` for :math:`x = 10^{23}`.
In `Deléglise et al, 2004 <Biblio.html#deleglise-dursat-roblot-04>`_, the authors computes the functions :math:`\pi(x,4,1)` and 
:math:`\pi(x,4,3)` that counts the number of prime numbers up to :math:`x` in the residue classes :math:`p \equiv 1 \mod 4` and
:math:`p \equiv 3 \mod 4`, respectively, for :math:`x = m \times 10^k`
for :math:`m = 1, 2, \dots, 9` and :math:`k = 10, 11, \dots, 20` (for :math:`k = 20`, only :math:`m = 1` was considered).



Paper  :cite:`Deleglise-Rivat96-1` computes the Chebyshev function :math:`\psi(x)` for :math:`x = m \times 10^k`
for :math:`m = 1, 2, \dots, 9` and :math:`k = 6, 7, \dots, 15` (for :math:`k = 15`, only :math:`m = 1` was considered).