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.

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

##### Keywords:

Riemann zeta-function; critical strip; exponential sum; average of short sums; Gauss sums; large sieve; mean value theorem##### References:

[1] | E. Bombieri - H. Iwaniec , Some mean-value theorems jor exponential sums , Ann. Scuola Norm. Sup. Pisa Cl. Sci. , 13 , no. 3 ( 1986 ), pp. 473 - 486 . Numdam | MR 881102 | Zbl 0615.10046 · Zbl 0615.10046 · numdam:ASNSP_1986_4_13_3_473_0 · eudml:83988 |

[2] | J.-M. Deshouillers - H. Iwaniec , Kloosterman sums and Fourier coefficients of cusp forms , Invent. Math. , 70 ( 1982 ), pp. 219 - 288 . MR 684172 | Zbl 0502.10021 · Zbl 0502.10021 · doi:10.1007/BF01390728 · eudml:142975 |

[3] | J.B. Friedlander - H. Iwaniec , On the distribution of the sequence n2\theta (mod 1) , to appear in the Canadian J. Math. Zbl 0625.10029 · Zbl 0625.10029 · doi:10.4153/CJM-1987-016-2 |

[4] | S.W. Graham - G. Kolesnik , One and two dimensional exponential sums , preprint 1984 (to be published in the Proceedings from the Conference on Number Theory Held at the OSU in July 1984 ). MR 1018377 | Zbl 0626.10034 · Zbl 0626.10034 |

[5] | G.H. Hardy , On certain definite integrals considered by Airy and by Stokes, Quart . J. Math. , 44 ( 1910 ), pp. 226 - 240 . JFM 41.0322.01 · JFM 41.0322.01 |

[6] | G. Kolesnik , On the method of exponent pairs , Acta Arith. , 55 ( 1985 ), pp. 115 - 143 . Article | MR 797257 | Zbl 0571.10036 · Zbl 0571.10036 · eudml:205960 |

[7] | E. Phillips , The zeta-function of Riemann: further developments of van der Corput’s method, Quart . J. Math. , 4 ( 1933 ), pp. 209 - 225 . Zbl 0007.29801 | JFM 59.0204.01 · Zbl 0007.29801 · www.emis.de |

[8] | R.A. Rankin , Van der Corput’s method and the theory of exponent pairs, Quart . J. Math. , 6 ( 1955 ), pp. 147 - 153 . MR 72170 | Zbl 0065.27802 · Zbl 0065.27802 · doi:10.1093/qmath/6.1.147 |

[9] | E.C. Titchmarsh , The Theory of the Riemann Zeta-Function , Oxford 1951 . MR 46485 | Zbl 0042.07901 · Zbl 0042.07901 |

[10] | J.D. Vaaler , Some extremal functions in Fourier analysis , Bull. Amer. Math. Soc. , 42 ( 2 ) ( 1985 ), pp. 183 - 216 . Article | MR 776471 | Zbl 0575.42003 · Zbl 0575.42003 · doi:10.1090/S0273-0979-1985-15349-2 · minidml.mathdoc.fr |

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