Rearrangements of the coefficients of entire Dirichlet series (Q2716752)

Let \(f(z)=\sum_{n=1}^\infty a_ne^{\lambda_nz}\) be an entire Dirichlet series with \(a_n\neq 0\), \(1<\lambda_1<\lambda_2<\dots\), \(\lambda_n\longrightarrow+\infty\), \(\limsup_{n\to+\infty}\frac{\lambda_n}{n}<+\infty\), \(\liminf_{n\to+\infty}(\lambda_{n+1}-\lambda_n)>0\). Let \(R(f)=\limsup_{n\to+\infty}\frac{\lambda_n\log\lambda_n}{\log\frac{1}{|a_n|}}\) denote the Ritt order of \(f\). Assume that \(0<R(f)<+\infty\). For any bijection \(\pi\:\mathbb N\longrightarrow\mathbb N\) put \(f_\pi(z):=\sum_{n=1}^\infty a_{\pi(n)}e^{\lambda_nz}\). The author proves that \(R(f_\pi)=R(f)\) iff \(\pi(n)=n+O(n)\). Moreover, he shows that the above condition is not sufficient for the equality of the Ritt type of \(f\) and \(f_\pi\).