The asymptotics of a new exponential sum

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