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A genus-zero canonical product for the zero set ${n^{2} : n \geq 1}$

Example

The genus-zero canonical product

$\prod_{n \geq 1} (1 - \frac{z}{n^{2}})$

converges normally on $C$ and has zeros exactly at the squares $n^{2}$ , each with multiplicity $1$ .

Facts & Assumptions

Given: The zero sequence $a_{n} = n^{2}$ .

[F1]

If $\sum ∣ a_{n} ∣^{- 1}$ converges, then the genus-zero canonical product $\prod E_{0} (z / a_{n})$ converges normally and has exactly the prescribed zeros (A canonical product converges when the $(p + 1)$ -power reciprocal sum converges).

Verification

1.1F1givenalgebra

Here $\sum_{n \geq 1} ∣ a_{n} ∣^{- 1} = \sum_{n \geq 1} 1 / n^{2}$ converges, and $E_{0} (w) = 1 - w$ , so [F1] applies with $p = 0$ .

2.1F1step 1.1∎

Therefore $\prod_{n \geq 1} (1 - z / n^{2})$ converges normally on $C$ and has zeros exactly at the points $n^{2}$ .

Depends on

A canonical product converges when the $(p+1)$-power reciprocal sum converges

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)

A genus-zero canonical product for the zero set {n2:n≥1}{n2:n≥1}