On a theorem of representation of harmonic functions (Q1896900)

A class of harmonic functions \(u(z)\) in \(|z|<1\) is studied, such that \(|u(z) |\leq AS\ln S\), where \(S=2/ (1-|z|)\) and \(A\) is a constant. It is shown that it is possible to write \(u(z)= u_1 (z)- u_2 (z)\) with \(u_1\) and \(u_2\) harmonic and bounded from above by \(MS/ \ln S\), \(M\) constant. The proof is very original -- the prime number function \(\pi (x)\) is involved! --, contains a couple of minor errors and demands much from the reader.