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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.

MSC:
 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