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

For each s ∈ ℕ define the constant θ_s with the following properties: if an entire function g(z) of type t(g) <θ_s satisfies g^(σ)(z) ∈ ℤ for σ = 0, 1,..., s - 1 and z = 0, 1, 2,..., then g is a polynomial; conversely, for any δ > 0 there exists an entire transcendental function g(z) satisfying the display conditin and t(g) <θ_s + δ. The result θ₁ = log 2 is known due to Hardy and Pólya. We provide the upper bound θ_s ≤ πs/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941).