Searching for an asymptotic to exp[0.5]

[tommy1729]

\( f(z) = C e^{Az + B} \prod_{j=1}^\infty (1-\frac{z}{a_j})e^{-\frac{z}{a_j}} \)

for some \( A,B,C \in \mathbb{C} \) where \( a_j \) is all the zeroes counted with multiplicity.

Your equation is overdetermined I think.

\( f(z) = C e^{Az} \prod_{j=1}^\infty (1-\frac{z}{a_j})e^{-\frac{z}{a_j}} \)

Removing \( B \) was trivial and the other variable \( A \) is correct because of the genus theory / theorem in combination with below :

This also implies, quite adequately that at least \( \sum_{j=1}^\infty |a_j|^{-(1+\epsilon)} < \infty \) for \( \epsilon > 0 \).

Well not from Hadamard alone , but from the asymptotes of the zero's yes.

And in combination with the genus theory we get the result above.

I did assume an upper bound on multiplicity though.

Thanks for the post James.

regards

tommy1729