Riemann Zeta Even Approximation
Real positive even zetaseriessumthroughNwithmonotoneintegral tail bounds, no analytic continuation or rounded partial as exactζ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
At an even real integer greater than one, the zeta reciprocal-power series converges.
At an even real integer greater than one, the zeta reciprocal-power series converges.1
This tool returns a finite partial sum with integral tail bounds rather than using analytic continuation or claiming an exact value.
When to use Riemann Zeta Even Approximation
This tool returns a finite partial sum with integral tail bounds rather than using analytic continuation or claiming an exact value.
How to use Riemann Zeta Even Approximation and how it works
- Prepare exponent, term budget, using the input definitions and matching units below.
- Enter the required fields and run Riemann Zeta Even Approximation.
- Read result.partialValue, result.integralLowerTail, result.integralUpperTail 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 |
|---|---|---|---|
| actualExponent | Integer reciprocal-power exponent with the parity and range specified below. | number | Required; integer 2–64; finite; other constraints below |
| actualTermBudget | Number of included terms under the stated series indexing convention. | number | Required; integer 1–100000; finite; other constraints below |
Require exponent as an integer 2–64 with the parity named by the tool, and term budget integer 1–100000. Sum k^(-s) through budget N. Return the partial value and positive omitted-tail integral bounds (N+1)^(1-s)/(s-1) and N^(1-s)/(s-1). Add the appropriate tail to the partial sum for the mathematical enclosure, subject to ordinary floating-point rounding; the returned bounds are not an interval-arithmetic rounding certificate.
partial=sum(k^-s,k=1..N); lowerTail=(N+1)^(1-s)/(s-1); upperTail=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.partialValue | Finite reciprocal-power sum; omitted tail is reported separately. | number |
| result.integralLowerTail | Mathematical lower bound on the positive omitted tail, excluding rounding certification. | number |
| result.integralUpperTail | Mathematical upper bound on the positive omitted tail, excluding rounding certification. | number |
{
"input": {
"actualExponent": 2,
"actualTermBudget": 10
},
"output": {
"result": {
"partialValue": 1.5497677311665408,
"integralLowerTail": 0.09090909090909091,
"integralUpperTail": 0.1
}
}
}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.
DLMF gives the convergent reciprocal-power series on the applicable real domain. This tool reports a partial sum and integral tail bounds rather than the infinite sum.2
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 Prime Zeta Approximation for a complementary operation: Prime zeta sums reciprocal powers over primes rather than over all positive integers.
References
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.
- Prime Zeta Approximation
Prime zeta sums reciprocal powers over primes rather than over all positive integers.