On the asymptotic connection between two exponential sums

  • Authors:

  • R. B. Paris

  • Affiliations:

  • Division of Mathematical Sciences, University of Abertay Dundee, Dundee DD1 1HG, UK

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2003

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Abstract

The relation between the exponential sums SN(x;p)= Σn=0N-1exp(πixnp) and T0 ≡ T0(x;N,p) = Σn=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) = Σn=1∞ exp(iθe-an)n-s for real θ and a 0.