Odd Zeta Approximation

Real positive odd 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 a real odd integer greater than one, the Riemann zeta function has a convergent reciprocal-power series.

At a real odd integer greater than one, the Riemann zeta function has a convergent reciprocal-power series.^[1]

This tool returns a finite partial sum and integral tail bounds rather than an exact infinite value.

When to use Odd Zeta Approximation

This tool returns a finite partial sum and integral tail bounds rather than an exact infinite value.

How to use Odd Zeta Approximation and how it works

Prepare exponent, term budget, using the input definitions and matching units below.
Enter the required fields and run Odd Zeta 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

json

{
  "input": {
    "actualExponent": 3,
    "actualTermBudget": 10
  },
  "output": {
    "result": {
      "partialValue": 1.197531985674193,
      "integralLowerTail": 0.004132231404958678,
      "integralUpperTail": 0.005
    }
  }
}

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 Riemann Zeta Even Approximation for a complementary operation: At an even real integer greater than one, the zeta reciprocal-power series converges.

Use Prime Zeta Approximation for a complementary operation: Prime zeta sums reciprocal powers over primes rather than over all positive integers.

References

Odd Zeta Approximation — tool.io
NIST DLMF: Riemann zeta definition and expansions — NIST

Similar or alternative tools

Riemann Zeta Even Approximation
At an even real integer greater than one, the zeta reciprocal-power series converges.
Prime Zeta Approximation
Prime zeta sums reciprocal powers over primes rather than over all positive integers.