PRIMECOUNT(1)                                                    PRIMECOUNT(1)

NAME
       primecount - count prime numbers

SYNOPSIS
       primecount x [options]

DESCRIPTION
       Count the number of primes less than or equal to x (<= 10^31) using
       fast implementations of the combinatorial prime counting function
       algorithms. By default primecount counts primes using Xavier Gourdon's
       algorithm which has a runtime complexity of O(x^(2/3) / log^2 x)
       operations and uses O(x^(2/3) * log^3 x) memory. primecount is
       multi-threaded, it uses all available CPU cores by default.

OPTIONS
       -d, --deleglise-rivat
           Count primes using the Deleglise-Rivat algorithm.

       -g, --gourdon
           Count primes using Xavier Gourdon's algorithm (default algorithm).

       -l, --legendre
           Count primes using Legendre's formula.

       --lehmer
           Count primes using Lehmer's formula.

       --lmo
           Count primes using the Lagarias-Miller-Odlyzko algorithm.

       -m, --meissel
           Count primes using Meissel's formula.

       --Li
           Approximate pi(x) using the Eulerian logarithmic integral: Li(x),
           with Li(x) = li(x) - li(2).

       --Li-inverse
           Approximate the nth prime using the inverse Eulerian logarithmic
           integral: Li^-1(x).

       -n, --nth-prime
           Calculate the nth prime.

       -p, --primesieve
           Count primes using the sieve of Eratosthenes.

       --phi X A
           phi(x, a) counts the numbers <= x that are not divisible by any of
           the first a primes.

       -R, --RiemannR
           Approximate pi(x) using the Riemann R function: R(x).

       --RiemannR-inverse
           Approximate the nth prime using the inverse Riemann R function:
           R^-1(x).

       -s, --status[=NUM]
           Show the computation progress e.g. 1%, 2%, 3%, ... Show NUM digits
           after the decimal point: --status=1 prints 99.9%.

       --test
           Run various correctness tests and exit.

       --time
           Print the time elapsed in seconds.

       -t, --threads=NUM
           Set the number of threads, 1 <= NUM <= CPU cores. By default
           primecount uses all available CPU cores.

       -v, --version
           Print version and license information.

       -h, --help
           Print this help menu.

ADVANCED OPTIONS FOR THE DELEGLISE-RIVAT ALGORITHM
       --P2
           Compute the 2nd partial sieve function.

       --S1
           Compute the ordinary leaves.

       --S2-trivial
           Compute the trivial special leaves.

       --S2-easy
           Compute the easy special leaves.

       --S2-hard
           Compute the hard special leaves.

   Tuning factor
       The alpha tuning factor mainly balances the computation of the S2_easy
       and S2_hard formulas. By increasing alpha the runtime of the S2_hard
       formula will usually decrease but the runtime of the S2_easy formula
       will increase. For large pi(x) computations with x >= 10^25 you can
       usually achieve a significant speedup by increasing alpha.

       The alpha tuning factor is also very useful for verifying pi(x)
       computations. You compute pi(x) twice but for the second computation
       you use a slightly different alpha factor. If the results of both pi(x)
       computations match then pi(x) has been verified successfully.

       -a, --alpha=NUM
           Set the alpha tuning factor: y = x^(1/3) * alpha, 1 <= alpha <=
           x^(1/6).

ADVANCED OPTIONS FOR XAVIER GOURDON'S ALGORITHM
       --AC
           Compute the A + C formulas.

       --B
           Compute the B formula.

       --D
           Compute the D formula.

       --Phi0
           Compute the Phi0 formula.

       --Sigma
           Compute the 7 Sigma formulas.

   Tuning factors
       The alpha_y and alpha_z tuning factors mainly balance the computation
       of the A, B, C and D formulas. When alpha_y is decreased but alpha_z is
       increased then the runtime of the B formula will increase but the
       runtime of the A, C and D formulas will decrease. For large pi(x)
       computations with x >= 10^25 you can usually achieve a significant
       speedup by decreasing alpha_y and increasing alpha_z. For convenience
       when you increase alpha_z using --alpha-z=NUM then alpha_y is
       automatically decreased.

       Both the alpha_y and alpha_z tuning factors are also very useful for
       verifying pi(x) computations. You compute pi(x) twice but for the
       second computation you use a slightly different alpha_y or alpha_z
       factor. If the results of both pi(x) computations match then pi(x) has
       been verified successfully.

       --alpha-y=NUM
           Set the alpha_y tuning factor: y = x^(1/3) * alpha_y, 1 <= alpha_y
           <= x^(1/6).

       --alpha-z=NUM
           Set the alpha_z tuning factor: z = y * alpha_z, 1 <= alpha_z <=
           x^(1/6).

EXAMPLES
       primecount 1000
           Count the primes <= 1000.

       primecount 1e17 --status
           Count the primes <= 10^17 and print status information.

       primecount 1e15 --threads 1 --time
           Count the primes <= 10^15 using a single thread and print the time
           elapsed.

HOMEPAGE
       https://github.com/kimwalisch/primecount

AUTHOR
       Kim Walisch <kim.walisch@gmail.com>

                                  02/17/2024                     PRIMECOUNT(1)