An asymptotic approximation for incomplete Gauss sums
The incomplete Gauss sum of order \(p>1\) is \[ S_m(2/N;p) = \sum_{j=0}^{m-1} \exp(2\pi ij^p/N). \] The graph traced by mapping the terms for \(m\leq N\) as a sequence of unit vectors in the complex plane exhibits spirals first described by \textit{D. H. Lehmer} (1976). The author obtains precise asymptotic estimates for the sum over the first spiral by expanding the terms of the sum: \[ S_m(x;p) = \sum_{k=0}^\infty{(\pi ix)^k\over k!} \sum_{j=0}^{m-1} j^{kp} = \sum_{k=0}^\infty {(i\theta)^k\over k!} \sum_{s=0}^{kp} {B_s\over s!} {(kp)!\over(kp+1-s)!} m^{1-s} \] where \(\theta=\pi xm^p\) and the \(B_s\) are the Bernoulli numbers. By inverting the double sum again: \[ S_m(x;p) \sim m _1F_1\left({1\over p};1+{1\over p};i\theta\right) +{1\over2}+ e^{i\theta}\left\{{1\over e^{2\pi i\chi}-1}-{1\over 2\pi i\chi}\right\} +e^{i\theta} \sum_{s=1}^\infty {B_{2s}\over(2s)!} {f_s(\theta;p)\over m^{2s-1}} \] where the \(f_s(\theta;p)\) are polynomials in \(\theta\) of degree \(2s-2\), provided \(\chi = p\theta/2\pi m < 1\). The expansion can even be given more precisely. When \(x=2/N\), the condition amounts to \(1\leq m<M_0\) where \(M_0=(N/p)^{1/(p-1)}\). In particular, when \(p=2\), the approximation for the classical incomplete Gauss sum is \[ S_m(2/N;2)\sim m_1F_1\left({1\over2};{3\over2};i\theta\right)+{1\over2} +e^{i\theta}\left\{{1\over e^{2\pi i\chi}-1}-{1\over 2\pi i\chi}\right\} + {\pi^2m\over15N^2} e^{i\theta} \] over the range \(1\leq m< N/2\). Several tables illustrate the estimate and show it is good. By way of comparison, previous estimates, for example that of \textit{R. Evans, M. Minei} and \textit{B. Yee} [J. Math. Anal. Appl. 281, No.~2, 454--476 (2003; Zbl 1028.11047)] obtained by application of the Poisson summation formula, hold only for large \(m<N/2\).
scientific article; zbMATH DE number 2175291
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