Potenzreihen mit fastgeraden Koeffizienten. (Power series with almost- even coefficients) (Q1262893)

An arithmetic function f: \({\mathbb{N}}\to {\mathbb{C}}\) is called even if it is a linear combination of the Ramanujan sums \[ c_ r(n)=\sum_{1\leq a\leq r,\quad (a,r)=1}e^{2\pi i(a/r)n}. \] The space of all even arithmetic functions is denoted by \({\mathcal B}\) and its closure in the space of all arithmetic functions with respect to the seminorm \[ \| f\|_ 2=\limsup ((1/x)\sum_{n\leq x}| f(n)|^ 2)^{1/2} \] is denoted by \({\mathfrak B}^ 2.\) Motivated by a result of \textit{L. Rubel} and \textit{K. Stolarsky} [Am. Math. Mon. 87, 371-376 (1980; Zbl 0467.30002)] characterizing those subsets \(A\subset {\mathbb{N}}\) for which the function \(\sum_{n\in A}z^ n/n!\) is bounded on the negative real axis, \textit{W. Schwarz} [Banach Cent. Publ. 17, 463-498 (1985; Zbl 0598.10049)] considered the function \(E_ f(z)=\sum^{\infty}_{n=1}f(n)z^ n/n!\) for \(f\in {\mathcal B}^ 2\), and gave necessary conditions on f in order for \(E_ f(z)\) to be bounded on the negative real axis. In the paper under review, the author proves analogous results under the weaker assumption that \(E_ f(z)\) satisfies \((*)\quad | E_ f(z)| \leq ce^{\vartheta | z|}\) for \(z<0\) with some constants \(c>0\) and \(0\leq \vartheta <1\). For example, he shows that a multiplicative function in \(f\in {\mathcal B}^ 2\) with non-zero mean value which satisfies (*) must be of the form \[ f(n)=\prod_{p| r}(1+\sum_{k\geq 1,\quad p^ k| (r,n)}p^ k\frac{a_{p^ k}- a_{p^{k+1}}}{a_ 1-a_ p}) \] for some integer r and suitable coefficients \(a_{p^ k}\).