Chebyshev Bias Calculator
FiniteobservedChebyshevprime-racebiasmod4, signed counts not forever ordering or logarithmic density. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
A prime race compares prime counts in specified residue classes over a finite interval.
A prime race compares prime counts in specified residue classes over a finite interval.1
This tool reports the observed modulo-four counts and their signed difference without assigning an infinite density.
When to use Chebyshev Bias Calculator
Use this tool to review the observed modulo-four counts and their signed difference without assigning an infinite density.
How to use Chebyshev Bias Calculator and how it works
- Prepare actualCutoff, actualFirstResidueModFour, actualSecondResidueModFour, with the units, conventions and assumptions below.
- Enter the required fields and run Chebyshev Bias Calculator.
- Read result.firstResiduePrimeCount, result.secondResiduePrimeCount, result.signedFirstMinusSecondCount using the output definitions below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualCutoff | Inclusive integer cutoff N for the finite operation; use the stated tool-specific range. | number | Required; integer 2–1000000; finite; other constraints below |
| actualFirstResidueModFour | First prime-race residue, exactly 1 or 3 modulo four. | number | Required; finite; other constraints below |
| actualSecondResidueModFour | Second residue, exactly 1 or 3 and different from the first. | number | Required; finite; other constraints below |
Require an integer prime cutoff from 2 to 1,000,000 and two distinct residues, each either 1 or 3 modulo 4. Count sieved primes in each residue class and return first-count minus second-count. The prime 2 belongs to neither class; reversing the residue order reverses the sign, and the finite count is not a claim about an infinite bias.
difference=count(p<=cutoff,p mod 4=first)-count(p<=cutoff,p mod 4=second) 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.firstResiduePrimeCount | Number of included primes in the first residue class. | number |
| result.secondResiduePrimeCount | Number of included primes in the second residue class. | number |
| result.signedFirstMinusSecondCount | Signed first count minus second count. | number |
{
"input": {
"actualCutoff": 11,
"actualFirstResidueModFour": 3,
"actualSecondResidueModFour": 1
},
"output": {
"result": {
"firstResiduePrimeCount": 3,
"secondResiduePrimeCount": 1,
"signedFirstMinusSecondCount": 2
}
}
}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 Euler Product Approximation for a complementary operation: The zeta Euler product combines factors indexed by primes on a convergent real-exponent domain.
Use Artin Constant Approximation for a complementary operation: Artin's constant has an Euler product over primes related to a primitive-root density convention.
References
-
Chebyshev Bias Calculator — tool.io
Similar or alternative tools
- Euler Product Approximation
The zeta Euler product combines factors indexed by primes on a convergent real-exponent domain.
- Artin Constant Approximation
Artin's constant has an Euler product over primes related to a primitive-root density convention.