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.

Description

Schnirelmann density is an infimum of membership proportions over all initial integer intervals.

Schnirelmann density is an infimum of membership proportions over all initial integer intervals. A finite observed prefix gives only a minimum over the intervals actually supplied.^[1]

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.

When to use Schnirelmann Density Calculator

Use this tool to 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.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Schnirelmann Density 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 membershipFlagsForIntegersOneToN using the field table and calculation rules, then run the tool.
Read finiteSchnirelmannPrefixMinimum, minimumAttainingPrefixLength, finiteWindowLength. 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
membershipFlagsForIntegersOneToN	Membership flags for integers one to n	list of number	Required

Supply 1–10,000 binary membership flags for integers 1 through N. Scan cumulative counts, retain the smallest prefix proportion and its earliest attaining length, and report N.

finite minimum = min[k=1..N] Σ[i=1..k]membership_i/k

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
finiteSchnirelmannPrefixMinimum	Finite schnirelmann prefix minimum	number
minimumAttainingPrefixLength	Minimum attaining prefix length	number
finiteWindowLength	Finite window length	number

json

{
  "input": {
    "membershipFlagsForIntegersOneToN": [
      1,
      0,
      1,
      0
    ]
  },
  "output": {
    "finiteSchnirelmannPrefixMinimum": 0.5,
    "minimumAttainingPrefixLength": 2,
    "finiteWindowLength": 4
  }
}

Limitations and assumptions

The finite minimum is an upper bound on the full infimum. It does not establish an infinite-set density.

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 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.

References

Schnirelmann Density 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.
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.