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on Erdős maximum term problem ($C_{51}$)

The statement

For any transcendental entire function $f(z)=\sum_{n\ge 0} a_n z^n$ , define .

M(r,f):=\max_{|z|=r}|f(z)|, \qquad \mu(r,f):=\max_{n\ge 0}|a_n|\,r^n.

Following [Er1961], define

\beta(f):=\liminf_{r\to\infty}\frac{\mu(r,f)}{M(r,f)}.

We define $C_{51} = B$ to be the supremum of $\beta(f)$ over all transcendental entire functions $f$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.