The Estimation of Complete Exponential Sums

Loxton, J. H. ; Vaughan, R. C.

Canadian mathematical bulletin, Tome 28 (1985) no. 4, pp. 440-454

Résumé

This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πi f(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.

Détail

DOI : 10.4153/CMB-1985-053-7

Mots-clés : 10G10, 12B05

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