Tommy's Gaussian method.

[tommy1729]

Let f(h(s)) = exp(f(s)) for 4 < Re(s) << Im(s).

Now I wonder where the singularities are for h(s).

since h(s) = inv.f(exp(f(s)))

I consider ( as a subquestion ) solutions v such that f ' (v) = 0. 

This matters for the " quality " of h(s) approximating s+1 when Im(s) =/= 0.

Although this might not be neccessary for a proof , it would help for the creation of multiple proofs.

In particular my view on a proof is there are basicly 2 kind ; " internal and external " ;

for 4 < Re(s) << Im(s) :
tet(s) = f(s) + error1(s)

or

for 4 < Re(s) << Im(s) :
tet(s) = f(s + error2(s))

where error2(s) is related to the questions about h(s).

I believe error2(s) implies error1(s) or in other words , the harder ( internal) proof implies the easier (external ) .


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arg(h(s)) is also very interesting and usefull to understand.

in particular  for 4 < Re(s) << Im(s).

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I think it also possible to prove (or decide ) the ( usual ) base change to be nowhere analytic since it strongly relates.

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regards

tommy1729