A note on entire dirichlet series of irregular growth

Year 1984, Volume: 33 , - , 01.01.1984

Satendra K. Vaısh

https://doi.org/10.1501/Commua1_0000000582

Abstract

F



or an entire Dirichlet series f(s) — S an exp(s Xn ), (s = a -H it, 0 < X: < X2 Xa < . . . 5 Üm sup ? = O), the order p(O < p < co ) and lower n -*°° Xn X(O < X < co ) are given by the limit superior and limit inferior of (log log M(a))/a, as a -► a» , respectively, where M(o) — sup ] ]f(a -f- it)}: -co < t < c©}. In this note w e derive a formula for lower proximate type T \ { — Hm inf (log M (G ))/e ° ^ [, O < Tx çy-JOO < oo, O < X < c o ,in terms of the coefficients a’n s and X’n s. Here X(o) is the lower proximate order or f(s).

Keywords

entire, dirichlet, irregular

References

• Communications, Series A1:Mathematics and Statistics