Geodesic
The Estimation of Complete Exponential Sums
Canadian mathematical bulletin, Tome 28 (1985) no. 4, pp. 440-454

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DOI

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.
DOI : 10.4153/CMB-1985-053-7
Mots-clés : 10G10, 12B05 
Loxton, J. H.; Vaughan, R. C. The Estimation of Complete Exponential Sums. Canadian mathematical bulletin, Tome 28 (1985) no. 4, pp. 440-454. doi: 10.4153/CMB-1985-053-7
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     title = {The {Estimation} of {Complete} {Exponential} {Sums}},
     journal = {Canadian mathematical bulletin},
     pages = {440--454},
     year = {1985},
     volume = {28},
     number = {4},
     doi = {10.4153/CMB-1985-053-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1985-053-7/}
}
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