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

The estimations of lower order (R) λ in terms of the sequences {a n } and {λ n } for an entire Dirichlet series f(s)= n=1 a n e sλn , have been obtained, namely :

λ=max{λnp}liminfpλnplogλnp-1log|anp|-1=max{λnp}liminfp(λnp-λnp-1)logλnp-1log|anp-1|anp|.

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

f(s)=n=1anesλn

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 ).

@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},
     address = {Gap},
     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/

[1] Alfred Gray and S.M. Shah, Asymptotic values of Holomorphic Functions of irregular growth, Bull. Amer. Math. Soc., 5, 71 (1965), 747-749. | MR 31 #3614 | Zbl 0147.06703

[2] Alfred Gray and S.M. Shah, Holomorphic Functions with Gap Power Series, Math. Zeit, 86 (1965), 375-394. | MR 33 #7501 | Zbl 0133.03501

[3] Alfred Gray and S.M. Shah, Holomorphic Functions with Gap Power Series, (II), Math. Anal. and Appl. 2, 13, (1966). | Zbl 0163.08802

[4] O.P. Juneja and Prem Singh, On the Lower order of an entire function defined by Dirichlet series, Math. Ann. 184 (1969), 25-29. | MR 40 #7448 | Zbl 0182.09702

[5] P.K. Kamthan, On entire functions represented by Dirichlet series (IV), Ann. Inst. Fourier, Grenoble, 16, 2 (1966), 209-223. | Numdam | MR 37 #1606 | Zbl 0145.08103

[6] Q.I. Rahman, On the lower order of entire functions defined by Dirichlet series. Quart. J. Math. Oxford (2), 7 (1956), 96-99. | MR 20 #5282 | Zbl 0074.29901