Potenzreihen, die einer Wachstumsbedingung genügen und additive Koeffizienten besitzen. (Power series satisfying a growth condition and having additive coefficients) (Q1262894)

The author gives a complete characterization of additive functions \(f\in {\mathcal B}^ 2\) satisfying (*) \(| \sum^{\infty}_{n=1}f(n)z^ n/n!| \leq ce^{\vartheta | z|}\) for \(z<0\) with some constants \(c\geq 0\) and \(0\leq \vartheta <1\). (For the definition of \({\mathcal B}^ 2\) see the preceding review.) He shows that (*) holds for an additive function \(f\in {\mathcal B}^ 2\) if and only if f is of the form \[ f(n)=\sum_{p\in U(\vartheta)}\sum_{k\geq 1,\quad p^ k| n}p^ k(b_{p^ k}-b_{p^{k+1}})+\sum_{k\geq 1,\quad 2^ k| n}2^ k(b_{2^ k}-b_{2^{k+1}}), \] where the set U(\(\vartheta)\) consists of all prime powers \(p^ k\leq 2\pi /\arccos \vartheta\) with the exception of 2, and the coefficients \(b_ n\) satisfy \(b_ n=0\) for \(n\not\in U(\vartheta)\).