Littlewood Violation Calculator
Finiteπ(x)−offset logarithmic integral under independent li(2) calibration/transformed quadrature, noglobalLittlewoodsignchangeproof/no huge first crossover search. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
A finite prime-counting discrepancy depends on the exact finite count and the chosen logarithmic-integral normalization.
A finite prime-counting discrepancy depends on the exact finite count and the chosen logarithmic-integral normalization.1
This tool combines a supplied offset with numerical quadrature without searching for a global first crossover or proving a sign-change theorem.
When to use Littlewood Violation Calculator
This tool combines a supplied offset with numerical quadrature without searching for a global first crossover or proving a sign-change theorem.
How to use Littlewood Violation Calculator and how it works
- Prepare cutoff, independent offset li at two, using the input definitions and matching units below.
- Enter the required fields and run Littlewood Violation Calculator.
- Read result.observedPi, result.conditionalLi, result.signedPiMinusConditionalLi, result.observedPositiveDiscrepancy 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 |
|---|---|---|---|
| actualCutoff | Inclusive upper integer cutoff for the finite counting or discrepancy operation. | number | Required; integer 2–1000000; finite; other constraints below |
| callerIndependentOffsetLiAtTwo | Finite independently verified value of the chosen logarithmic-integral convention at x=2. | number | Required; finite; other constraints below |
Require cutoff integer 2–1000000 and finite caller-calibrated offset li at two. Count primes through cutoff exactly. Integrate exp(t)/t from ln(2) to ln(cutoff) using 2048-panel Simpson quadrature and add the caller offset. Return pi, conditional li, their signed discrepancy and whether it is positive. Offset convention and quadrature uncertainty remain external, and no first crossover or global sign theorem is established.
conditionalLi=callerLiAtTwo+Simpson(exp(t)/t,ln(2),ln(N),2048); discrepancy=pi(N)-conditionalLi 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.observedPi | Exact finite number of primes no greater than the cutoff. | number |
| result.conditionalLi | Caller offset plus the finite numerical logarithmic-integral quadrature. | number |
| result.signedPiMinusConditionalLi | Finite prime count minus the conditional logarithmic integral. | number |
| result.observedPositiveDiscrepancy | True only when the finite signed discrepancy is strictly positive. | boolean |
{
"input": {
"actualCutoff": 10,
"callerIndependentOffsetLiAtTwo": 1.045163780117493
},
"output": {
"result": {
"observedPi": 4,
"conditionalLi": 6.165599504787351,
"signedPiMinusConditionalLi": -2.165599504787351,
"observedPositiveDiscrepancy": false
}
}
}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 Skewes Number Approximation for a complementary operation: Historical Skewes bounds concern an upper bound on a prime-counting crossover under specified assumptions.
Use Prime Number Theorem Approximation for a complementary operation: The prime number theorem describes asymptotic prime-counting behavior rather than an exact finite identity.
References
-
Littlewood Violation Calculator — tool.io
Similar or alternative tools
- Skewes Number Approximation
Historical Skewes bounds concern an upper bound on a prime-counting crossover under specified assumptions.
- Prime Number Theorem Approximation
The prime number theorem describes asymptotic prime-counting behavior rather than an exact finite identity.