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.

--double-check
    Recompute pi(x) with alternative alpha tuning factor(s) to verify the first result. This redundancy helps guard against potential bugs in primecount: if an error exists, it is highly unlikely that both pi(x) computations would produce the same (incorrect) result.

-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).

--RiemannR-psi
    Approximate pi(x) using a refined Riemann R approximation based on R(psi(x)). Compared to --RiemannR, this option first approximates Chebyshev’s psi(x) using Riemann’s explicit formula with the first 512 zeta zero gammas and then evaluates R(psi(x)). In practice this usually estimates pi(x) more accurately than plain R(x).

--RiemannR-psi-inverse
    Approximate the nth prime using the inverse of the refined R(psi(x)) approximation. Compared to --RiemannR-inverse, this option inverts the psi-corrected approximation instead of plain R(x). In practice this usually estimates the nth prime more accurately than 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 usually decreases, while the runtime of the S2_easy formula increases. By default, primecount automatically selects an alpha tuning factor that is likely to provide the best performance for the user’s input.

The alpha tuning factor is also very useful for double checking 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 and alpha_z is increased, the runtime of the B formula increases, while the runtime of the A, C and D formulas decrease. By default, primecount automatically selects alpha_y and alpha_z tuning factors that are likely to provide the best performance for the user’s input.

Both the alpha_y and alpha_z tuning factors are also very useful for double checking pi(x) computations. You compute pi(x) twice but for the second computation you use slightly different alpha_y and/or alpha_z tuning factors. 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).

DOUBLE CHECKING PI(X) COMPUTATIONS

For large pi(x) computations that run over several weeks or months, it is important to verify such computations to protect against miscalculations due to hardware errors and/or bugs in primecount.

To verify a pi(x) computation, you compute pi(x) twice. But for the second run you use the --double-check option. The --double-check option enables the use of alternative alpha tuning factor(s), ensuring that all internal bounds in the second computation differ slightly from the first run. This redundancy helps guard against potential bugs in primecount: if an error exists, it is highly unlikely that both pi(x) computations would produce the same (incorrect) result.

primecount 1e20 --status
    First pi(x) computation using default alpha tuning factor(s).

primecount 1e20 --status --double-check
    Second pi(x) computation using alternative alpha tuning factor(s).

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>