So we're taking,
\(
G(-ks) = \int_1^\infty e^{-x}x^{-1-ks}\,dx\\
\)
which has decay to zero as \( \Re(s) \to \infty \). However, you then take,
\(
\Omega_{j=1}^\infty e^{z+G(-k(s-j))}\bullet z\\
\)
But, \( G(-k(s-j)) = G(jk-ks) \to \infty \) as \( j \to \infty \). As in,
\(
\sum_{j=1}^\infty e^{z+G(-k(s-j))} = \infty\\
\)
So I'm not sure how you are planning to derive convergence of this infinite composition.
\(
G(-ks) = \int_1^\infty e^{-x}x^{-1-ks}\,dx\\
\)
which has decay to zero as \( \Re(s) \to \infty \). However, you then take,
\(
\Omega_{j=1}^\infty e^{z+G(-k(s-j))}\bullet z\\
\)
But, \( G(-k(s-j)) = G(jk-ks) \to \infty \) as \( j \to \infty \). As in,
\(
\sum_{j=1}^\infty e^{z+G(-k(s-j))} = \infty\\
\)
So I'm not sure how you are planning to derive convergence of this infinite composition.
