In this note we prove a theorem which gives us Information as to how the functions log I§ (a) and log (a) grow relative to each other as a ^00. Theorem. Let f(s) be an entire function represented by a Dirichlet series, then lim a ^00 log Ig (<’) log J8,k (o) X ■! (1 + k/X) (X 0) (X = 0) where X = lim CT-^OO log log Ig (a) a 1. In the usual notation, 00 f(s) = s 1 Sn e®^-”, (s = (j it), 0 A, (n 1) lim n -*oo 00 is an entire function in the sense that the Dirichlet series represen- ting it, is ahsolutely convergent for ali finite s and possesses where 0 lim (7-4-00 log log M(g) (7 X, 00, and M(ct) have their usual meanings.