Some relations between arguments of Gauss sums.
The author considers the classical exponential sums \(S(a x^{k},p) = \sum _{j \pmod p} \exp(2\pi \imath j^{k}/p\) where \(p\) is a prime, \(p \equiv 1 \pmod k\). Let \(g\) be a primive root \(\pmod p.\) One knows from the theory of Gauss sums that \(S(g^{t} x^{k},p)\) can be expressed as a trigonometric sum in \(t\) where the phases are arguments of Gauss sums formed with character of order \(k\). There is no absolute order on the set of phases but there is one for each \(p\) conferred by the choice of primitive root. The number of distinct phases is \([(k-1)/2]\) where we exclude for the purposes the trivial and quadratic characters, the latter in the case where \(k\) is even. In an incautious moment \textit{H. L. Montgomery, R. C. Vaughan} and \textit{T. D. Wooley} suggested in [Math. Proc. Camb. Philos. Soc. 118, No. 1, 21--33 (1995; Zbl 0835.11032), pp. 24--25] that with a choice of primitve roots for the different primes \(p\) the set of phases would be dense in the set of all possible phases. They also remarked that nothing was known about the linear combinations of two phases. In this paper the author remarks that one of the Davenport-Hasse relations leads to explicit formulas for certain linear combinations of three phases. These show that the suggestion of Montgomery, Vaughan and Wooley is not tenable in the original form. The reviewer would like to take this opportunity to clarify a point that has been left somewhat unclear both in the paper under review and in the paper cited above. In [Proc. Lond. Math. Soc., III. Ser. 54, 193--215 (1987; Zbl 0612.10030)] the reviewer showed that certain Gauss sums at prime arguments defined in the ring of integers of a cyclotomic field are uniformly distributed. These Gauss sums are, however, not the same as the ones defined over the rationals. They differ by a factor which typically is the inverse of the Legendre symbol, at such a prime \(\pi\), of \(p/\pi\) (where \(p\) is the norm of \(\pi\)). In the cubic case this is known and is a consequence of cubic reciprocity; the biquadratic case is more recent (1979) and due to C. R. Matthews [see \textit{K. Ireland} and \textit{M. Rosen}, A Classical Introduction to Modern Number Theory (1990; Zbl 0712.11001), p. 136, Ex. 30]. Unfortunately no analogue of these results is known in higher cases and this is the desideratum for the application of the reviewer's results to exponential sums of classical type.
scientific article; zbMATH DE number 2074702