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Lectures on a method in the theory of exponential sums. (English) Zbl 0671.10031
Tata Institute of Fundamental Research. Berlin etc.: Springer-Verlag. viii, 134 p. DM 20.00 (1987).
One of the essential foundations of van der Corput’s method of estimating exponential sums of type \(\sum_{a<n\leq b}e^{2\pi if(n)}\) is to transform the sum into another one by means of Poisson’s summation formula. The aim of this book is to estimate sums of the form \[ \sum_{a<n\leq b}b(n) e^{2\pi if(n)}, \] where \(b(n)\) is the number of divisors of \(n\) or the Fourier coefficients of a cusp form. For this purpose the author develops a method, where Poisson’s formula is replaced by Voronoi’s summation formula.
Chapter 1 deals with the Dirichlet series \(\sum^{\infty}_{n=1}b(n) e^{2\pi inr} n^{-s}\), where \(r\) is a rational number. Functional equations and an analytic behaviour of these functions are investigated. Also approximate formulas of the Voronoi type for the exponential sums \(\sum_{n\leq x}b(n) e^{2\pi inr}\) are derived. The main results of this chapter are summation formulas of Voronoi type.
In Chapter 2 exponential integrals are estimated by means of the saddle point method.
The basic results of Chapter 3 are transformation formulas for exponential sums \[ \sum_{M_ 1<m\leq M_ 2}b(m)g(m)e^{2\pi if(m)}, \] where \(M_ 1<M_ 2\leq 2M_ 1\) and \(g(m)\) is a suitable function, by combining the summation formulas of Chapter 1 and the estimations of exponential integrals of Chapter 2.
In Chapter 4 the results of the preceding chapter are applied to Dirichlet polynomials \(\sum d(m) m^{-1/2-it}\), \(\sum a(m) m^{- \kappa /2-it}\) with the divisor function \(d(m)\) and the Fourier coefficients \(a(m)\) of a cusp form of weight \(\kappa\) for the full modular group. The chapter is finished with proofs of Heath-Brown’s estimation of the twelfth-power moment of \(\zeta (1/2+it)\) and of the sixth-power moment \[ \int^{T}_{0}| \phi (\kappa /2+it)|^ 6 \,dt \ll T^{2+\varepsilon}, \] where \(\phi(s)=\sum^{\infty}_{n=1}a(n)n^{-s}\).
The book is clearly written and is a welcome contribution to the theory of exponential sums.

11L03 Trigonometric and exponential sums, general
11L40 Estimates on character sums
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
30B50 Dirichlet series, exponential series and other series in one complex variable