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.

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

-v, --version

-h, --help

# 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.