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Exponential sums and the Riemann zeta function. IV. (English) Zbl 0803.11046
This extensive work is the fourth part of a series [Part I, Proc. Lond. Math. Soc., III. Ser. 57, 1-24 (1988; Zbl 0644.10027), Part II, J. Lond. Math. Soc., II. Ser. 39, 385-404 (1989; Zbl 0678.10027), Part III, Proc. Lond. Math. Soc., III. Ser. 62, 449-468 (1991; Zbl 0734.11044)] by M. N. Huxley and N. Watt, N. Watt, and M. N. Huxley and G. Kolesnik, respectively. It concerns estimation of the exponential sum $S= \sum_{M\leq m\leq M_ 2} e(TF( {\textstyle {m\over M}})) \tag{1}$ for an arbitrary smooth function $$F$$, where $$e(x)= e^{2\pi ix}$$ and $$M_ 2< 2M<T$$. The method, originally due to Bombieri-Iwaniec, consists of six steps. The present paper treats the sixth step, namely counting the number of “resonances” between the so- called minor arcs, given by the approximate coincidence of certain vectors. Three lengthy theorems are proved, which furnish bounds for the sum $$S$$. As an application of the results one has $\zeta \bigl( {\textstyle {1\over 2}}+ iT\bigr) \ll_ \varepsilon T^{{{89} \over {570}}+ \varepsilon}, \qquad {\textstyle {{89} \over {570}}}= 0.15614035\dots\;,$ which is hitherto the sharpest unconditional bound. This paper, together with the first three parts of the series, represents an important contribution to the study of exponential sums of type (1).
 11L07 Estimates on exponential sums 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11L40 Estimates on character sums