On theorems of Gelfond and Selberg concerning integral-valued entire functions

For each s ∈ N define the constant s with the following properties: if an entire function g(z) of type t (g)< s satisfies g (z) ∈ Z for = 0, 1, . . . , s − 1 and z= 0, 1, 2, . . . , then g is a polynomial; conversely, for any > 0 there exists an entire transcendental function g(z) satisfying the display conditin and t (g)< s + . The result 1 = log 2 is known due to Hardy and Pólya. We provide the upper bound s s/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941). © 2004 Elsevier Inc. All rights reserved.