Could be tetration if this integral converges

[tommy1729]

JmsNxn,

I am a little suspicious of this method. In particular, I'm not sure the integral

\( I = \int_{0}^{\infty} \vartheta(-x) x^{z-1} dx \), \( 0 < z < 1 \)

converges.

I totally agree. Although I added a minus sign to it, which I assume was a typo.

So lets think about \( \vartheta(-x) \).

Since the Taylor coefficients of \( \vartheta(-x) \) decay EXTREMELY FAST , I consider this as a function that is well approximated by a polynomial for a long time.
( many remainder theorems for Taylor series imply this )

This means the main behaviour of this \( \vartheta(-x) \) is like \( (-1)^n a_n x^n \) where n increases slowly with x.

This implies that \( \vartheta(-x) \) is not bounded by a polynomial and also that \( \vartheta(-x) \) = 0 infinitely often.

Therefore the integral diverges.

Even if we consider taking the limit of x going to +oo as the limit of the sequence x_i with \( \vartheta(-x_i) \) = 0.

regards

tommy1729