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On the order of $$\zeta(+it)$$. (English) Zbl 0615.10047
(The review of ibid. 473-486 is included.) In these most interesting papers, the new estimate $\zeta (1/2+it)\ll t^{9/56+\epsilon}$ is obtained. Note that $$9/56=0.16071...$$, whilest the best known exponent was previously $$139/858=0.162004...$$, due to G. Kolesnik [Acta Arith. 45, 115-143 (1985; Zbl 0571.10036)]. The arguments of the authors are essentially more arithmetic than those traditionally used in this problem. But the starting point is, as usual, the estimation of the exponential sum $$\sum_{m\sim M}m^{it}$$, where $$M\ll t^{1/2}$$. Consider, more generally, the sum $$\sum e(f(m))$$; only in the last step of the proof it is necessary to specify $$f(x)=(t/2\pi)\log x$$. As in the methods of Weyl and van der Corput, the average of short sums $$| \sum_{n}e(f(m+n)|$$ is estimated when m runs over a long interval, but now this sum is analyzed and transformed before averaging. First $$f(m+n)$$ is approximated by its cubic Taylor polynomial, and introducing rational approximations for its second and first power coefficients, one is led to ”perturbed” Gauss sums $$\sum_{n\sim N}e(\mu n^ 3+\frac{a}{c}n^ 2+\frac{b}{c}n)$$. This can be transformed by use of Poisson’s summation formula and elementary properties of Gauss sums to another sum which is more amenable to averaging, done by (a form of) the large sieve. Then two purely arithmetic problems arise.
The first of these is estimating how often the inequalities $$\| \bar a/c-\bar a_ 1/c_ 1\| <\Delta_ 1$$ and $$| ac-a_ 1c_ 1| <\Delta_ 2AC$$ can be simultaneously fulfilled for a, $$a_ 1\sim A$$ and c, $$c_ 1\sim C$$. Here $$\| \cdot \|$$ means the distance from the nearest integer, and $$a\bar a\equiv 1$$ (mod c). The result is $$\ll AC+\Delta_ 2(A^ 2+C^ 2)+\Delta_ 1(\Delta_ 1+\Delta_ 2)A^ 2C^ 2$$ (actually there is an extra factor $$(AC)^{\epsilon}$$, but is has been shown by M. N. Huxley and N. Watt that this can be removed).
The second problem is estimating the number of integral eight-tuples $$(h_ 1,...,h_ 8)$$ with $$h_ j\sim H$$ such that $$\sum^{4}_{1}(h^ 2_ j-j^ 2_{j+4})\ll 1,\sum^{4}_{1}(h_ j-h_{j+4})\ll 1$$, and $$\sum^{4}_{1}(h_ j^{3/2}- h^{3/2}_{j+4})\ll H^{1/2}$$. This is solved in the second paper as a corollary of a mean value estimate for the exponential sum $S(\alpha,\beta,x;N)=\sum_{n\sim N}e(\alpha n^ 2+\beta n+xf(n/N)),$ where $$f(t)=t^{\kappa}$$ ($$\kappa\neq 0,1,2)$$ or $$f(t)=\log t$$. The result $\int^{1}_{0}\int^{1}_{0}\int^{X}_{-X}| S|^ 2 d\alpha d\beta dx\ll (X+N)N^{4+\epsilon},$ and the number of solutions in the above system is $$\ll H^{4+\epsilon}$$. In the proof of the mean value theorem, Poisson’s summation formula is applied in an ingenious iterative way.
The ideas of these papers work for more general sums if the lemma involving a, c, $$a_ 1$$, $$c_ 1$$ is appropriately generalized. This has been done by Huxley and Watt in a paper to appear. Also, in a forthcoming paper, H. Iwaniec and C. J. Mozzochi are going to prove the estimate $$\Delta (x)\ll x^{7/22+\epsilon}$$ in Dirichlet’s divisor problem by similar methods.
Reviewer: M.Jutila

MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11L40 Estimates on character sums
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References:
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