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 logarithmic integral.

       --Li-inverse
           Approximate the nth prime using 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.

       --Ri
           Approximate pi(x) using the Riemann R function.

       --Ri-inverse
           Approximate the nth prime using Ri^-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>

                                            07/17/2023                              PRIMECOUNT(1)