Montgomery Pair Correlation Calculator
Compute finite supplied-ordinate pair-gap histogram normalized by caller mean spacing, counting unordered i<j pairs once; does not verify ordinates are zeta zeros or prove Montgomery asymptotic pair correlation.
Description
Pair-correlation sampling counts spacing between supplied ordinates relative to a declared mean spacing.
Pair-correlation sampling counts spacing between supplied ordinates relative to a declared mean spacing. Finite pair counts do not establish an asymptotic distribution.1
Compute finite supplied-ordinate pair-gap histogram normalized by caller mean spacing, counting unordered i<j pairs once; does not verify ordinates are zeta zeros or prove Montgomery asymptotic pair correlation.
When to use Montgomery Pair Correlation Calculator
Use this tool to compute finite supplied-ordinate pair-gap histogram normalized by caller mean spacing, counting unordered i<j pairs once; does not verify ordinates are zeta zeros or prove Montgomery asymptotic pair correlation.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Montgomery Pair Correlation Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter orderedPositiveZeroOrdinates, callerMeanSpacing, normalizedGapLowerBound, normalizedGapUpperBound using the field table and calculation rules, then run the tool.
- Read observedUnorderedPairCountInBand, pairsPerSuppliedOrdinate, totalComparedPairs. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| orderedPositiveZeroOrdinates | Ordered positive zero ordinates | list of number | Required |
| callerMeanSpacing | Caller mean spacing | number | Required |
| normalizedGapLowerBound | Normalized gap lower bound | number | Required |
| normalizedGapUpperBound | Normalized gap upper bound | number | Required |
Supply 2–1000 strictly increasing positive ordinates, positive mean spacing and a nonnegative positive-width band. Count each unordered pair once in the half-open interval [lower,upper); return count, count per supplied ordinate and total pairs compared.
Each sequence checked for increasing order must contain at least two strictly increasing values.
normalized gap = (ordinate_j−ordinate_i)/mean spacing, i<j Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| observedUnorderedPairCountInBand | Observed unordered pair count in band | number |
| pairsPerSuppliedOrdinate | Pairs per supplied ordinate | number |
| totalComparedPairs | Total compared pairs | number |
{
"input": {
"orderedPositiveZeroOrdinates": [
10,
12,
15
],
"callerMeanSpacing": 2,
"normalizedGapLowerBound": 0,
"normalizedGapUpperBound": 2
},
"output": {
"observedUnorderedPairCountInBand": 2,
"pairsPerSuppliedOrdinate": 0.6666666666666666,
"totalComparedPairs": 3
}
}Limitations and assumptions
The tool does not verify that ordinates are zeta zeros or prove Montgomery's asymptotic conjecture.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Natural Density Calculator: Compute density at a finite declared integer cutoff from complete binary membership; not global prefix infimum or proof that infinite natural-density limit exists. For a complementary task, use Schnirelmann Density Calculator: Compute minimum observed prefix density over explicit membership of integers 1..N; finite-prefix minimum is an upper bound on global Schnirelmann infimum, not certified infinite-set density.
References
-
Montgomery Pair Correlation Calculator — tool.io
Similar or alternative tools
- Natural Density Calculator
Compute density at a finite declared integer cutoff from complete binary membership; not global prefix infimum or proof that infinite natural-density limit exists.
- Schnirelmann Density Calculator
Compute minimum observed prefix density over explicit membership of integers 1..N; finite-prefix minimum is an upper bound on global Schnirelmann infimum, not certified infinite-set density.