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Spring 2014 - Problem 8

Weierstrass factorization theorem

Construct a non-constant entire function $f\p{z}$ such that the zeroes of $f$ are simple and coincide with the set of all (positive) natural numbers.

Solution.

By the Weierstrass factorization theorem,

f\p{z} = \prod_{n=1}^\infty E_1\p{\frac{z}{n}}

works, where

E_1\p{z} = e^z\p{1 - z}.

We have the bound $\abs{1 - E_1\p{z}} \leq \abs{z}^2$ for $\abs{z} \leq 1$ . Thus, on compact balls $\cl{B\p{0, R}}$ , if $N \in \N$ is such that $N - 1 < R \leq N$ , we have $\abs{\frac{z}{n}} \leq \frac{R}{N} \leq 1$ for $n \geq N$ . Thus,

\sum_{n=N}^\infty \abs{1 - E_1\p{z}} \leq \sum_{n=N}^\infty \p{\frac{R}{n}}^2 = R^2 \sum_{n=N}^\infty \frac{1}{n^2} < \infty,

so the product converges locally uniformly on compact sets, and so $f$ is an entire function with the specified zeroes.