PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The convolution $1 * λ$ detects perfect squares

Statement

For every positive integer $n$ ,

(1 * λ) (n) = {\begin{cases} 1, & n is a perfect square, \\ 0, & n is not a perfect square . \end{cases}

Facts & Assumptions

Given: A positive integer $n$ .

Proof

technique · direct

1.1

Write $n = \prod_{i < r} p_{i}^{e_{i}}$ by For $n \geq 1$ and any injective list $p : r \to Z$ of primes containing every prime divisor of $n$ , one has $n = \prod_{i < r} p_{i}^{v_{p_{i}} (n)}$ ; the exponents are determined by $n$ , and $v_{q} (n) = 0$ for every prime $q$ outside the list. Every positive divisor of $n$ has the form $\prod_{i < r} p_{i}^{j_{i}}$ with $0 \leq j_{i} \leq e_{i}$ , so Dirichlet convolution of arithmetic functions and Liouville's function give $(1 * λ) (n) = \sum_{d ∣ n} λ (d) = \prod_{i < r} (\sum_{j = 0}^{e_{i}} (- 1)^{j})$ , where the factorization of the finite sum uses Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule.

givenconstruct

2.1

For each $i$ , the alternating sum $\sum_{j = 0}^{e_{i}} (- 1)^{j}$ equals $1$ when $e_{i}$ is even and $0$ when $e_{i}$ is odd. Therefore the product in step 1.1 is $1$ exactly when every exponent $e_{i}$ is even, and otherwise it is $0$ .

step 1.1algebra

3.1

By canonical factorization, every exponent $e_{i}$ is even exactly when $n$ is a perfect square. So step 2.1 is precisely the claimed square-indicator formula.

step 2.1∎

Depends on

Used by

Working out the square-indicator convolution for prime powers Example

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)
Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 2.9 (standard reference, not scraped)

The convolution 1∗λ1∗λ detects perfect squares