Zeros of Sections of Exponential Sums

We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the nth section of the exponential sum into “genuine zeros,” which approach, as n → ∞, the zeros of the exponential sum, and “spurious zeros,” which go to infinity as n → ∞. We show that the spurious zeros, after scaling down by the factor of n, approach a “rosette,” a finite collection of curves on the complex plane, resembling the rosette. We derive also the large n asymptotics of the “transitional zeros,” the intermediate zeros between genuine and spurious ones. Our results give an extension to the classical results of Szegö about the large n asymptotics of zeros of sections of the exponential, sine, and cosine functions.