The Growth of an Entire Function and its Dirichlet Coefficients and Exponents

Let F be an entire function represented by a (generalized) Dirichlet series where the coefficients { d n } and exponents { n } ( n = 1, 2, …) are sequences of complex numbers. We introduce a modified (R)-order 𝜌 and modified (R)-type σ and we obtain an estimate for | d n | when n is sufficiently large in terms of 𝜌 , σ and n . Other estimates relating 𝜌 and σ to { n } and { d n } are also obtained.