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

Jain, P. K.; Jain, D. R.

On the lower order

(R)

of an entire Dirichlet series

Jain, P. K. ; Jain, D. R.

Annales de l'Institut Fourier, Tome 24 (1974), p. 123-129 / Harvested from Numdam

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Résumé
Abstract

Des estimations $λ$ de l’ordre inférieur $(R)$ d’une série de Dirichlet

f (s) = \sum_{n = 1}^{\infty} a_{n} e^{s λ n}

ont été obtenues en fonction des suites ${a_{n}}$ et ${λ_{n}}$ .

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 ).

The estimations of lower order $(R)$ $λ$ in terms of the sequences ${a_{n}}$ and ${λ_{n}}$ for an entire Dirichlet series $f (s) = \sum_{n = 1}^{\infty} a_{n} e^{s λ n}$ , have been obtained, namely :

\begin{matrix} λ & = max_{{λ_{n_{p}}}} lim inf_{p \to \infty} \frac{λ_{n_{p}} log λ_{n_{p - 1}}}{log | a_{n_{p}} |^{- 1}} \\ = max_{{λ_{n_{p}}}} lim inf_{p \to \infty} \frac{(λ_{n_{p}} - λ_{n_{p - 1}}) log λ_{n_{p - 1}}}{log | a_{n_{p - 1}} | a_{n_{p}} |} . \end{matrix}

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 ).

Publié le : 1974-01-01

MR 50 #7520 | Zbl 0273.30021

DOI : https://doi.org/10.5802/aif.494

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     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},
     volume = {24},
     year = {1974},
     pages = {123-129},
     doi = {10.5802/aif.494},
     mrnumber = {50 \#7520},
     zbl = {0273.30021},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 24 (1974) pp. 123-129. doi : 10.5802/aif.494. http://gdmltest.u-ga.fr/item/AIF_1974__24_1_123_0/

Bibliographie

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