Meissel-Mertens Constant Approximation
Meissel–MertensprimeharmonicconstantB1≈0.2614972128476428. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Description
The Meissel-Mertens constant appears in the limiting correction to the sum of reciprocal primes.
The Meissel-Mertens constant appears in the limiting correction to the sum of reciprocal primes.1
This tool rounds a stored approximation without computing a prime harmonic sum or certifying extra precision.
When to use Meissel-Mertens Constant Approximation
Obtain the named stored numerical approximation rounded for a display or calculation.
How to use Meissel-Mertens Constant Approximation and how it works
- Choose an integer decimal precision from 0 to 12.
- Enter decimalPlaces and run the tool.
- Read result.value as a number; trailing zeros are not preserved.
- Use a separately certified high-precision source when the stored approximation cannot meet the required numerical accuracy.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| decimalPlaces | Integer display precision from 0 to 12; numerical output does not preserve trailing zeros. | number | Required; integer 0–1000000; integer 0–12; finite; other constraints below |
Require decimalPlaces as an integer from 0 to 12. Round the stored binary64 approximation 0.2614972128476428 with toFixed, then convert its decimal string back to a number. Return the numeric value, so trailing decimal zeros are not preserved. The underlying constant is not recomputed and added decimal places do not provide arbitrary precision or a rigorous error certificate.
value=Number(storedApproximation.toFixed(decimalPlaces)) 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.value | Numeric model result on the ratio, score, constant or density basis explained above. | number |
{
"input": {
"decimalPlaces": 6
},
"output": {
"result": {
"value": 0.261497
}
}
}Limitations and assumptions
Rounding is applied to a stored binary64 approximation. It does not recompute the constant, provide rigorous error bounds or increase the precision of the stored value.
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 Porter Constant Approximation for a complementary operation: Porter's constant occurs in an average-cost expansion for the Euclidean algorithm under a fixed convention.
Use Hafner-Sarnak-McCurley Constant Approximation for a complementary operation: The Hafner-Sarnak-McCurley constant concerns a limiting density of coprime determinants under a specified random integer-matrix convention.
References
-
Meissel-Mertens Constant Approximation — tool.io
Similar or alternative tools
- Porter Constant Approximation
Porter's constant occurs in an average-cost expansion for the Euclidean algorithm under a fixed convention.
- Hafner-Sarnak-McCurley Constant Approximation
The Hafner-Sarnak-McCurley constant concerns a limiting density of coprime determinants under a specified random integer-matrix convention.