False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29

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FALSE: every zero sequence admits a genus-zero canonical product

Statement

Every discrete zero sequence in $C$ admits a genus-zero canonical product.

Facts & Assumptions

Given: The sequence $a_{n} = n$ for $n \geq 1$ .

[F1]

The genus-zero elementary factor is $E_{0} (w) = 1 - w$ (Weierstrass elementary factors).

[F2]

The harmonic series $\sum_{n \geq 1} 1 / n$ diverges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ).

[F3]

For nonnegative reals $p_{n}$ , the product $\prod (1 + p_{n})$ converges if and only if $\sum p_{n}$ converges (For $p_{k} \geq 0$ the product $\prod (1 + p_{k})$ converges iff $\sum p_{k}$ converges, with $1 + \sum_{k < n} p_{k} \leq \prod_{k < n} (1 + p_{k}) \leq 1 / (1 - \sum_{k < n} p_{k})$ when $\sum_{k < n} p_{k} < 1$ ; for $0 \leq p_{k} < 1$ the product $\prod (1 - p_{k})$ converges iff $\sum p_{k}$ converges and its partial products tend to $0$ otherwise; and $\sum ∣ p_{k} ∣$ convergent implies $\prod (1 + p_{k})$ convergent).

Refutation

1.1

The sequence $a_{n} = n$ has no finite accumulation point, so it is a legitimate zero sequence.

given

2.1

For this sequence the genus-zero canonical product is $\prod_{n \geq 1} E_{0} (z / n) = \prod_{n \geq 1} (1 - \frac{z}{n})$ by [F1]. Evaluating at $z = - 1$ gives the positive-factor product $\prod_{n \geq 1} (1 + \frac{1}{n}) .$

F1step 1.1algebra

3.1

The series $\sum_{n \geq 1} 1 / n$ diverges by [F2], so [F3] implies that the product in step 2.1 does not converge. Therefore the genus-zero canonical product for $a_{n} = n$ fails even pointwise at $z = - 1$ , and the statement is false.

F2F3step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 5 Infinite products (standard reference, not scraped)