Exponential sums with monomials
Exponential sums in several variables \(\sum... \sum e(f(m_ 1,m_ 2,...,m_ j))\), where f is a smooth function, are important in analytic number theory. The paper deals with the estimation of such sums, where f is a monomial function \[ f(m_ 1,m_ 2,...,m_ j)=xm_ 1^{\alpha_ 1}m_ 2^{\alpha_ 2}... m_ j^{\alpha_ j}. \] The method is founded on an inequality for bilinear forms \(\sum \sum \phi_ r\psi_ se(x_ ry_ s)\) with finite sequences of real numbers \(x_ r\) and \(y_ s\) and complex numbers \(\phi_ r\) and \(\psi_ s\), which was proved by \textit{E. Bombieri} and \textit{H. Iwaniec} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 449-472 (1986; Zbl 0615.10047)]. From this inequality it is derived an estimation in case of \(j=4\) and \[ f(m_ 1,...,m_ 4)=x(m_ 1/M_ 1)^{\alpha_ 1}... (m_ 4/M_ 4)^{\alpha_ 4}, \] where the \(m_ i\) in the sum run over intervals of order \(M_ i\). The result is useful in the range \(1\ll x\ll M_ 1... M_ 4\) and will be trivial for larger x. In order to obtain nontrivial results in the latter case the method is combined with Weyl's shifts and Poisson's summation. Finally, applications are given for special sums which occur in sieve problems for short intervals.
scientific article; zbMATH DE number 4125487
scientific article