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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. --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 inte- gral: 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 af- ter 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) com- putations. 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 in- creased, 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 per- formance 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 fac- tors. 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> 04/16/2026 PRIMECOUNT(1)
NAME | SYNOPSIS | DESCRIPTION | OPTIONS | ADVANCED OPTIONS FOR THE DELEGLISE-RIVAT ALGORITHM | ADVANCED OPTIONS FOR XAVIER GOURDON'S ALGORITHM | DOUBLE CHECKING PI(X) COMPUTATIONS | EXAMPLES | HOMEPAGE | AUTHOR
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