As a note on a similar technique you are applying Mike but trying to keep the vibe much more fractional calculus'y (since it's what I am familiar with) We will try the following function:

We want fairly small

We know that for because

So that .

This function should be smaller then tetration at natural values.

We would get the entire expression for by Lemma 3 of my paper:

Now F(z) will be susceptible to alot of the techniques I have in my belt involving fractional calculus. This Idea just popped into my head but I'm thinking working with a function like this will pull down the imaginary behaviour and pull down the real behaviour.

We also note that

. Which again will be more obvious if you look at the paper, but it basically follows because:

We want fairly small

We know that for because

So that .

This function should be smaller then tetration at natural values.

We would get the entire expression for by Lemma 3 of my paper:

Now F(z) will be susceptible to alot of the techniques I have in my belt involving fractional calculus. This Idea just popped into my head but I'm thinking working with a function like this will pull down the imaginary behaviour and pull down the real behaviour.

We also note that

. Which again will be more obvious if you look at the paper, but it basically follows because: