"Fast Method for Computing the Number of Primes Less Than a Given Limit” describes three processes used during the course of calculation. In the first part of the paper the author proves:
\[
ϕ
(
x
,
a
)
=
ϕ
(
x
,
1
)
−
ϕ
(
x
p
2
,
1
)
−
ϕ
(
x
p
3
,
2
)
−
…
−
ϕ
(
x
p
a
,
a
−
1
)
\phi (x,a) = \phi (x,1) - \phi ({\frac {x}{{{p_2}}},1}) - \phi ({\frac {x}{{{p_3}}},2}) - \ldots - \phi ({\frac {x}{{{p_a}}},a - 1})
\]
where
ϕ
(
x
,
a
)
\phi (x,a)
represents the number of numbers less than or equal to x and not divisible by the first “a” primes. This identity is used to evaluate the formula
π
(
x
)
=
ϕ
(
x
,
a
)
+
a
−
1
\pi (x) = \phi (x,a) + a - 1
,
a
+
1
>
π
(
x
)
a + 1 > \pi (\sqrt x )
where resulting terms of the form
ϕ
(
x
′
,
a
′
)
\phi (x’,a’)
are broken down still further by the previously described method, or numerically evaluated using one or both of two other identities, the choice being dependent on
x
′
x’
and
a
′
a’
. Following the paper is a table of calculations made using this process which gives the values of
π
(
x
)
\pi (x)
for x at intervals of 10 million up to 1000 million, along with the Riemann and the Chebyshev approximations for
π
(
x
)
\pi (x)
and the amount they deviate from the true count.