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1 results

  • The Estimation of Complete Exponential Sums

  • J. H. Loxton, R. C. Vaughan
  • Journal:
    Canadian Mathematical Bulletin / Volume 28 / Issue 4 / 01 December 1985
    Published online by Cambridge University Press:
    20 November 2018, pp. 440-454
    Print publication:
    01 December 1985
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  • This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.