On the lower order $\left(R\right)$ of an entire Dirichlet series

Jain, P. K. ; Jain, D. R.

Annales de l'Institut Fourier, Tome 24 (1974) no. 1, pp. 123-129

Abstract (VO)

The estimations of lower order $\left(R\right)$ $\lambda$ in terms of the sequences $\left\{a_n\right\}$ and $\left\{{\lambda }_n\right\}$ for an entire Dirichlet series $f\left(s\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{s\lambda n}$, have been obtained, namely :

$$\begin{array}{cc}\hfill \lambda & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{{\lambda }_{n_p}log{\lambda }_{n_{p-1}}}{log|a_{n_p}{|}^{-1}}\hfill \\ & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{({\lambda }_{n_p}-{\lambda }_{n_{p-1}})log{\lambda }_{n_{p-1}}}{log\left|a_{n_{p-1}}\right|a_{n_p}|}.\hfill \end{array}$$

One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

Résumé

Des estimations $\lambda$ de l’ordre inférieur $\left(R\right)$ d’une série de Dirichlet

$$f\left(s\right)=\sum _{n=1}^{\infty }{a}_n{e}^{s\lambda n}$$

ont été obtenues en fonction des suites $\left\{a_n\right\}$ et $\left\{{\lambda }_n\right\}$.

Ces estimations améliorent considérablement celles obtenues précédemment par Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)), Juneja et Singh (Math. Ann., 184 (1969), 25-29 ).

Détail

DOI : 10.5802/aif.494