The asymptotics of a new exponential sum

The absolutely convergent exponential sum S p ( θ ; m ) = ∑ n = 0 ∞ ′ exp ( − n / m − i θ e − p n / m ) θ > 0 , p > 0 is studied for m → + ∞ and fixed p when the parameter θ is allowed to become large such that θ / m remains finite. This situation corresponds, in general, to the trace in the complex plane of the partial sums of S p ( θ ; m ) consisting of a multiple spiral structure. Numerical results are presented to illustrate the accuracy of the expansion.