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2 Added the sum and zeta(zeta(s))=1 case

About $\zeta(s)=1$. Your "illusion" about zeros close to $\Re(s)=54$ well might be caused by working with insufficient precision. Probably zeta is so close to $1$ your precision believes it is exactly $1$.

Does the "illusion" disappear if you work with more precision?

My results with sage/mpmath:

With precision 16 decimal digits find a lot of zeros close to $\Re(s)=77$.

With precision 40 digits the zeros move to $\Re(s)=82$.

With precision 100 can't find large zeros fast, only with $\Re(s)<2$.

The sum for zeta for $\Re(s)>1$ explains why you get close to $1$ for large $\Re(s)$.

For $\zeta(\zeta(s))=1$ the solution again appear to be related to insufficient precision. The solutions I found are with large $\Re(\zeta(s))$, again tending to $1$.

1

About $\zeta(s)=1$. Your "illusion" about zeros close to $\Re(s)=54$ well might be caused by working with insufficient precision. Probably zeta is so close to $1$ your precision believes it is exactly $1$.

Does the "illusion" disappear if you work with more precision?

My results with sage/mpmath:

With precision 16 decimal digits find a lot of zeros close to $\Re(s)=77$.

With precision 40 digits the zeros move to $\Re(s)=82$.

With precision 100 can't find large zeros, only with $\Re(s)<2$.