Landau-Ramanujan Constant Approximation
FiniteLandau–RamanujanEulerproduct2^−1/2Πp≡3mod4(1−p^−2)^−1/2, not exact infinite constant. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
The Landau-Ramanujan constant has an Euler product over primes congruent to three modulo four.
The Landau-Ramanujan constant has an Euler product over primes congruent to three modulo four.1
This tool evaluates the finite product through a supplied prime cutoff without claiming the exact infinite constant.
When to use Landau-Ramanujan Constant Approximation
This tool evaluates the finite product through a supplied prime cutoff without claiming the exact infinite constant.
How to use Landau-Ramanujan Constant Approximation and how it works
- Prepare prime cutoff, using the input definitions and matching units below.
- Enter the required fields and run Landau-Ramanujan Constant Approximation.
- Read result.truncatedLandauRamanujanConstant using the units, signs and ordering explained below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualPrimeCutoff | Inclusive upper bound on prime values included in the finite sum or product. | number | Required; integer 2–1000000; finite; other constraints below |
Require prime cutoff integer 2–1000000. Sieve primes and keep p congruent to 3 modulo 4. Sum -ln(2)/2-ln(1-1/p^2)/2 in log space and exponentiate for the finite product. Return the truncated Landau-Ramanujan value without a tail certificate.
truncatedConstant=2^(-1/2)*product((1-p^-2)^(-1/2),prime p<=N and p mod 4=3) Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.truncatedLandauRamanujanConstant | Finite Landau-Ramanujan product under the stated branch and truncation; no additional numerical certificate. | number |
{
"input": {
"actualPrimeCutoff": 7
},
"output": {
"result": {
"truncatedLandauRamanujanConstant": 0.7577722283113838
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
Alternative or Complementary methods
Use Meissel-Mertens Constant Approximation for a complementary operation: The Meissel-Mertens constant appears in the limiting correction to the sum of reciprocal primes.
Use Prime Zeta Approximation for a complementary operation: Prime zeta sums reciprocal powers over primes rather than over all positive integers.
References
-
Landau-Ramanujan Constant Approximation — tool.io
Similar or alternative tools
- Meissel-Mertens Constant Approximation
The Meissel-Mertens constant appears in the limiting correction to the sum of reciprocal primes.
- Prime Zeta Approximation
Prime zeta sums reciprocal powers over primes rather than over all positive integers.