Euler Totient Sum Approximation
Exact finiteΣn=1…Nφ(n), N≤1e6, BigInt decimal output, not mislabeledasymptotic. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
The totient summatory function adds the counts of positive integers coprime to each index through a finite cutoff.
The totient summatory function adds the counts of positive integers coprime to each index through a finite cutoff.1
This tool computes that finite sum exactly and returns a decimal string rather than an asymptotic estimate.
When to use Euler Totient Sum Approximation
Use this tool to compute that finite sum exactly and returns a decimal string rather than an asymptotic estimate.
How to use Euler Totient Sum Approximation and how it works
- Prepare actualCutoff, with the units, conventions and assumptions below.
- Enter the required fields and run Euler Totient Sum Approximation.
- Read result.exactTotientSummatory using the output definitions below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualCutoff | Inclusive integer cutoff N for the finite operation; use the stated tool-specific range. | number | Required; finite; other constraints below |
Require an integer cutoff N from 0 to 1,000,000. Build an Euler totient sieve and sum phi(1) through phi(N) with BigInt arithmetic, returning an exact decimal string. N=0 gives "0". Despite the title, this is a finite exact sum rather than an asymptotic estimate.
value=sum_{n=1..N} phi(n), encoded as decimal string 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.exactTotientSummatory | Exact finite totient sum as a decimal integer string. | string |
{
"input": {
"actualCutoff": 5
},
"output": {
"result": {
"exactTotientSummatory": "10"
}
}
}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 Divisor Summatory Calculator for a complementary operation: The divisor-count summatory function totals the number of positive divisors across integers through a cutoff.
Use Divisor Sum Approximation for a complementary operation: A sum of divisor sums weights each divisor by its value across the finite prefix.
References
-
Euler Totient Sum Approximation — tool.io
Similar or alternative tools
- Divisor Summatory Calculator
The divisor-count summatory function totals the number of positive divisors across integers through a cutoff.
- Divisor Sum Approximation
A sum of divisor sums weights each divisor by its value across the finite prefix.