construct entire function Can anyone please explain to me how to use Weierstrass product theorem to construct entire function? 
If I want to construct entire function with simple zeros on the positive real axis at the points $\sqrt n , n\ge1$ and double zeros on the imaginary axis at the points $ \pm i \sqrt n , n\ge1$ with no other zeros.
There is a trick to the Weierstrass product. If $a_k$, $k\geq 1$ is a sequence tending to zero then $$\prod_{k\geq 1} (1-a_k z)$$ need not be convergent. You may, however, recover convergence by multiplying by a factor that has no zeros. You start by looking at $$1/(1-a_k z) = \exp( -\ln (1-a_k z))=\exp(\sum_{j\geq 1} \frac{(a_kz)^j}{j})$$ but you truncate the sum in the exponential to a finite order $p_k$.
By choose suitable $p_k$ (which may tend to infinity) the product:$$\prod_{k\geq 1} (1-a_k z) \exp\left(\sum_{j= 1}^{p_k} \frac{(a_kz)^j}{j}\right) $$
is an entire function.
let be the sets of zeros $ x_i $ of a function $ f(x_i) $ then the entire function can be constructed in the form

$$ f(x)= \prod _{n=0}^{\infty}(1-\frac{x}{x_{i}}) $$

then this $ f(x) $ is an entire function, see the product

$$ \prod _{n=0}^{\infty}x_{i}$$

must be zeta regularizable