Honestly, I'm super thankful that my ideas have affected you so much, Tommy. But I'd be much more interested in a discussion of the family of solutions.
Let's take:
\[
T[t](s+1) = e^{t(s) T(s)}\\
\]
Where the functions: \(t\) form a quasi space of functions such that:
\[
\sum_{j=1}^\infty |t(s-j)| < \infty\\
\]
And:
\[
T[t](s) = \Omega_{j=1}^\infty e^{t(s-j)z}\,\bullet z\\
\]
But the "addition" on this space is \(t_0 \oplus t_1 = \frac{t_0(s) + t_1(s)}{2}\), and \(t(\infty) = 1\). Both mine and your methods, and all of your methods belong to this space. I'm pretty fucking confident nothing is going to work for \(b=e\), it might for others. But base \(e\) is just so fucking volatile. And I'm confident both our methods fail at holomorphy in \(\mathbb{C}\).I'd be very surprised at this point if any \(t\) produces holomorphic functions. And if we do find one, it'll be partitioned as \(\Im s \to \pm \infty\) that \(t \to L^{\pm}\). Or something like that.... We can only find a representation for Kneser in this manner. We're not going to break things....
Let's take:
\[
T[t](s+1) = e^{t(s) T(s)}\\
\]
Where the functions: \(t\) form a quasi space of functions such that:
\[
\sum_{j=1}^\infty |t(s-j)| < \infty\\
\]
And:
\[
T[t](s) = \Omega_{j=1}^\infty e^{t(s-j)z}\,\bullet z\\
\]
But the "addition" on this space is \(t_0 \oplus t_1 = \frac{t_0(s) + t_1(s)}{2}\), and \(t(\infty) = 1\). Both mine and your methods, and all of your methods belong to this space. I'm pretty fucking confident nothing is going to work for \(b=e\), it might for others. But base \(e\) is just so fucking volatile. And I'm confident both our methods fail at holomorphy in \(\mathbb{C}\).I'd be very surprised at this point if any \(t\) produces holomorphic functions. And if we do find one, it'll be partitioned as \(\Im s \to \pm \infty\) that \(t \to L^{\pm}\). Or something like that.... We can only find a representation for Kneser in this manner. We're not going to break things....
