Vol. 18, No. 2, 1966

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Vol. 286: 1  2
Vol. 285: 1  2
Vol. 284: 1  2
Vol. 283: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Subscriptions
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
On an entire function of an entire function defined by Dirichlet series

K. N. Srivastava

Vol. 18 (1966), No. 2, 379–383
Abstract

In this note we prove the following theorem which seems to exhibit an essential property of the order (R) of entire function defined by Dirichlet series.

Theorem If h(s) and g(s) are entire functions defined by Dirichlet series and g(log h(s)) is an entire function of finite order (R), then there are only two possible cases: either

(a) the internal function h(s) is a Dirichlet polynomial and the external function g(s) is of finite order (R); or

(b) the internal function h(s) is of finite order (R) and the external function g(s) is of order zero.

Here h(s) and g(s) are entire functions defined by the Dirichlet series

      ∑∞    λns
h(s) =   ane
n=1

      ∞∑    ns
g(s) =    bne  ,
n=0

satisfying the relations

 l.u.b.|h(σ+ it)| = H (σ),
−∞ <t<∞

−l∞.u<.tb<.∞|g(σ+ it)| = G (σ),

for any real value of (in particular, every Dirichlet series absolutely convergent in the whole plane will have this property).

Mathematical Subject Classification
Primary: 30.56
Secondary: 30.24
Milestones
Received: 11 March 1965
Published: 1 August 1966
Authors
K. N. Srivastava