On the lower order $\left(R\right)$ of an entire Dirichlet series
Annales de l'Institut Fourier, Volume 24 (1974) no. 1, p. 123-129

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 :

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

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 ).

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 ).

@article{AIF_1974__24_1_123_0,
author = {Jain, P. K. and Jain, D. R.},
title = {On the lower order $(R)$ of an entire Dirichlet series},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {24},
number = {1},
year = {1974},
pages = {123-129},
doi = {10.5802/aif.494},
zbl = {0273.30021},
mrnumber = {50 \#7520},
language = {en},
url = {http://www.numdam.org/item/AIF_1974__24_1_123_0}
}

Jain, P. K.; Jain, D. R. On the lower order $(R)$ of an entire Dirichlet series. Annales de l'Institut Fourier, Volume 24 (1974) no. 1, pp. 123-129. doi : 10.5802/aif.494. http://www.numdam.org/item/AIF_1974__24_1_123_0/

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