Sequences of maximal terms and central exponents of derivatives of Dirichlet series

M. N. Sheremeta

Matematičeskie zametki, Tome 63 (1998) no. 3, pp. 457-467

Résumé

For the Dirichlet series corresponding to a function $F$ with positive exponents increasing to $\infty$ and with abscissa of absolute convergence $A\in(-\infty,+\infty]$, it is proved that the sequences $\bigl(\mu(\sigma,F^{(m)})\bigr)$ of maximal terms and $\bigl(\Lambda(\sigma,F^{(m)})\bigr)$ of central exponents are nondecreasing to $\infty$ as $m\to\infty$ for any given $\sigma$, and $$ \varlimsup_{m\to\infty}\frac{\ln\mu(\sigma,F^{(m)})}{m\ln m}\le1 \quad\text{and}\quad \varlimsup_{m\to\infty}\frac{\ln\Lambda(\sigma,F^{(m)})}{\ln m}\le1. $$ Necessary and sufficient conditions for putting the equality sign and replacing $\varlimsup$ by $\lim$ in these relations are given.