Prime Zeta Approximation
Real prime zeta sum over actual primes≤cutoff, s>1, conservativeALL-integer integral tail upper bound; not sum overall integersζ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
Prime zeta sums reciprocal powers over primes rather than over all positive integers.
Prime zeta sums reciprocal powers over primes rather than over all positive integers.1
This tool returns a finite prime sum and a conservative all-integer integral tail upper bound for a real exponent above one.
When to use Prime Zeta Approximation
This tool returns a finite prime sum and a conservative all-integer integral tail upper bound for a real exponent above one.
How to use Prime Zeta Approximation and how it works
- Prepare real exponent, prime cutoff, using the input definitions and matching units below.
- Enter the required fields and run Prime Zeta Approximation.
- Read result.partialPrimeZeta, result.integerIntegralUpperTail 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 |
|---|---|---|---|
| actualRealExponent | Actual Real Exponent. Use the units and matching context stated in the calculation below. | number | Required; finite; other constraints below |
| actualPrimeCutoff | Inclusive upper bound on prime values included in the finite sum or product. | number | Required; integer 2–1000000; finite; other constraints below |
Require real exponent s>1 and prime cutoff integer 2–1000000. Sieve all primes through cutoff and sum p^(-s). Return that partial prime zeta sum and cutoff^(1-s)/(s-1) as a conservative omitted-tail upper bound using all integers. This is not analytic continuation or a rounding-certified enclosure.
partial=sum(p^-s,prime p<=N); tailUpper=N^(1-s)/(s-1) 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.partialPrimeZeta | Finite prime zeta sum under the stated branch and truncation; no additional numerical certificate. | number |
| result.integerIntegralUpperTail | Conservative all-integer integral upper bound on the omitted prime-zeta tail. | number |
{
"input": {
"actualRealExponent": 2,
"actualPrimeCutoff": 7
},
"output": {
"result": {
"partialPrimeZeta": 0.42151927437641723,
"integerIntegralUpperTail": 0.14285714285714285
}
}
}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 Odd Zeta Approximation for a complementary operation: At a real odd integer greater than one, the Riemann zeta function has a convergent reciprocal-power series.
Use Riemann Zeta Even Approximation for a complementary operation: At an even real integer greater than one, the zeta reciprocal-power series converges.
References
-
Prime Zeta Approximation — tool.io
Similar or alternative tools
- Odd Zeta Approximation
At a real odd integer greater than one, the Riemann zeta function has a convergent reciprocal-power series.
- Riemann Zeta Even Approximation
At an even real integer greater than one, the zeta reciprocal-power series converges.