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Exponential sums and lattice points. II. (English) Zbl 0820.11060
The classical problems of improving the error terms in the asymptotic formulas for
(i) the number of lattice points inside a circle of radius \(\sqrt {x}\), and
(ii) The sum \(\sum_{n\leq x} d(n)\) where \(d(n)\) is the divisor function,
have attracted the attention of many of the most famous number theorists throughout our century. From about 1920 to the middle of the eighties, only tiny improvements were obtained, every ten or twenty years. Then C. J. Mozzochi and H. Iwaniec [J. Number Theory 29, 60-93 (1988; Zbl 0644.10031)] followed by M. N. Huxley [Proc. Lond. Math. Soc., III. Ser. 60, 471-502 (1990; Zbl 0696.10045)] succeeded in achieving a breakthrough, by a new and ingenious method.
In the present manuscript, the author further refines the deep analysis of his further work. He obtains the (“new-record”-) order term \(O(x^{23/ 73} (\log x)^ C)\) for both of the classical problems.
More generally, the author proves the analogous upper bound for sums \[ \sum_{M< m\leq M_ 2} \psi \Bigl( {\textstyle {T\over M}} F\Bigl( {\textstyle {m\over M}} \Bigr) \Bigr) \] (where \(\psi(t)= t- [t]- {1\over 2}\), \(M_ 2\leq 2M\)), under comparatively mild assumptions about the derivatives of \(F\) on \([1, 2]\).
Another “highlight” of the present paper is an improved asymptotic formula for the number of lattice points in a large (“blown up”) planar domain whose boundary is a piecewise smooth curve with finite nonzero curvature throughout. Here the estimate is again of the same accuracy as in the classic problems.
The results of this manuscript are certainly applicable to quite a number of other specific lattice point problems. As an application outside number theory (in the narrow sense), the author himself establishes some very interesting numerical quadrature formulas with sharp error terms.
The analysis which is necessary to obtain these deep results is one of the hardest and most complicated ones which probably occur in the contemporary mathematical literature. Keeping this in mind, one can say that the author actually has done very well in his attempt to present the paper in a readable form.
Reviewer: W.G.Nowak (Wien)

11P21 Lattice points in specified regions
11L07 Estimates on exponential sums
65D32 Numerical quadrature and cubature formulas
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