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

Jain, P. K., and Jain, D. R.. "On the lower order $(R)$ of an entire Dirichlet series." Annales de l'institut Fourier 24.1 (1974): 123-129. <http://eudml.org/doc/74157>.

@article{Jain1974,
abstract = {The estimations of lower order $(R)$$\lambda $ in terms of the sequences $\lbrace a_n\rbrace $ and $\lbrace \lambda _n\rbrace $ for an entire Dirichlet series $f(s)=\sum ^\infty _\{n=1\}a_ne^\{s\lambda n\}$, have been obtained, namely :\begin\{align\} \lambda & =\max \_\{ \lbrace \lambda \_\{n\_p\}\rbrace \}\lim \inf \_\{p\rightarrow \infty \} \{ \lambda \_\{n\_p\}\log \lambda \_\{n\_\{p-1\}\} \over \log \vert a\_\{n\_p\}\vert ^\{-1\}\}\\ & =\max \_\{ \lbrace \lambda \_\{n\_p\}\rbrace \}\lim \inf \_\{p\rightarrow \infty \} \{(\lambda \_\{n\_p\}-\lambda \_\{n\_\{p-1\}\}) \log \lambda \_\{n\_\{p-1\}\}\over \log \vert a\_\{n\_\{p-1\}\}\vert a\_\{n\_p\}\vert \}.\end\{align\}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 ).},
author = {Jain, P. K., Jain, D. R.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {1},
pages = {123-129},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the lower order $(R)$ of an entire Dirichlet series},
url = {http://eudml.org/doc/74157},
volume = {24},
year = {1974},
}

TY - JOUR
AU - Jain, P. K.
AU - Jain, D. R.
TI - On the lower order $(R)$ of an entire Dirichlet series
JO - Annales de l'institut Fourier
PY - 1974
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 1
SP - 123
EP - 129
AB - The estimations of lower order $(R)$$\lambda $ in terms of the sequences $\lbrace a_n\rbrace $ and $\lbrace \lambda _n\rbrace $ for an entire Dirichlet series $f(s)=\sum ^\infty _{n=1}a_ne^{s\lambda n}$, have been obtained, namely :\begin{align} \lambda & =\max _{ \lbrace \lambda _{n_p}\rbrace }\lim \inf _{p\rightarrow \infty } { \lambda _{n_p}\log \lambda _{n_{p-1}} \over \log \vert a_{n_p}\vert ^{-1}}\\ & =\max _{ \lbrace \lambda _{n_p}\rbrace }\lim \inf _{p\rightarrow \infty } {(\lambda _{n_p}-\lambda _{n_{p-1}}) \log \lambda _{n_{p-1}}\over \log \vert a_{n_{p-1}}\vert a_{n_p}\vert }.\end{align}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 ).
LA - eng
UR - http://eudml.org/doc/74157
ER -