On the asymptotic connection between two exponential sums

The relation between the exponential sums SN(x;p)=∑n=0N−1 exp(πixnp) and T0≡T0(x;N,p)=∑n=1∞ e−n/Nexp(πixNpe−pn/N), where x⩾0 and p>0, is investigated. It is demonstrated that there is an asymptotic connection as N→∞ which is found numerically to be valid provided the variable x satisfies the restriction xNp=o(N) when p>1. The sum T0 is shown to be associated with a zeta function defined by Z(s)=∑n=1∞ exp(iθe−an)n−s for real θ and a>0.