On the Zeros of the Sections of the Exponential Function

The Eneström-Kakeya theorem (the absolute value of any zero of a0+a1x+· · ·+akx, ai ∈ R is at most max( a0 a1 , a1 a2 , . . . , ak−1 ak ) ) implies |νj| ≤ k . It was shown by Szegö [8] that the numbers νj k cluster around the simple closed curve Γ = {z : |ze1−z| = 1, |z| ≤ 1} as k → ∞ and conversely each point of the Szegö curve is a limit point of the normalized zeros. Moreover Szegö [8] and also Dieudonné [4] found that as k → ∞ the proportion of the normalized zeros which cluster along a given arc of Γ is asymptotic to 1 2π ∆ arg ze1−z as z moves on the arc. Buckholtz [1] added that νj k always lies in the exterior of Γ within