On the divisor and circle problems.

*(English)*Zbl 0644.10031This remarkable paper contains the estimate \(\Delta (x)\ll x^{7/22+\epsilon}\), and \(P(x)\ll x^{7/22+\epsilon}\), where as usual
\[
\Delta (x)=\sum_{n\leq x}d(n)-x(\log x+2\gamma -1),\quad P(x)=\sum_{n\leq x}r(n)-\pi x
\]
denote the error terms in the divisor and circle problem, respectively. The exponent \(7/22=0.3181818...\) improves the previously best known exponent \(139/429=0.324009324...,\) which is due to G. Kolesnik [Acta Arith. 45, 115-143 (1985; Zbl 0571.10036)]. Although the improvement on the value of the exponent is substantial, even more important is the wealth of ideas that are used in the paper. The details of the proof are given only for \(\Delta\) (x), since some slight modifications of the authors’ arguments also work for the closely related circle problem (for this and a discussion of the history of estimates for \(\Delta(x)\) and P(x) see Ch. 13 of the reviewer’s book [“Riemann zeta-function” (Wiley 1985; Zbl 0556.10026)]).

Instead of treating \(\Delta(x)\) by (double) exponential sums via Voronoi’s formula, as was done by several generations of number theorists from E. C. Titchmarsh to G. Kolesnik, the authors adopt a new approach. They start from the elementary formula \[ (1)\quad \Delta (x)=- 2\sum_{n\leq \sqrt{x}}\psi (x/n)+0(1),\quad \psi (t)=t-[t]-1/2 \] and use a variety of techniques. These include approximation by Farey points and ideas from the Weyl and van der Corput treatment of exponential sums, such as Poisson summation. The (technically quite complicated) paper draws on ideas from the pioneering work of E. Bombieri and the first author [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 473-486 (1986; Zbl 0615.10046) and ibid. Ser. 13, 449-472 (1986; Zbl 0615.10047)] in which they proved \(\zeta (1/2+it)=O(t^{9/56+\epsilon}).\) Note that \(7/22<2\cdot (9/56)=0.3214285...,\) so that in a certain sense the methods of the present paper are even more successful, because one generally expects the exponent that one obtains in the divisor problem to be twice the exponent in the problem of the estimation of \(\zeta (1/2+it)\) (see the reviewer’s monograph for a discussion of this), since both problems can be reduced to the estimation of similar exponential sums. However, the present work uses intrinsically (1), whose analogue is not known to hold for \(\zeta (1/2+it),\) and at present it does not seem possible to derive \(\zeta (1/2+it)=O(t^{7/44+\epsilon})\) by the methods of the paper under review.

An important role in the proof is played by “incomplete theta-series” \[ \theta (\alpha,\beta)=\sum_{n}g(n) e(\alpha n+\beta n^ 2)\quad (e(x)=e^{2\pi ix}), \] where \(\alpha\), \(\beta\) are reals and g(n) is a smooth real function, compactly supported. In bounding \(\theta\) (\(\alpha\),\(\beta)\) the authors are led to the evaluation of the integral \[ \int^{\infty}_{0}x^{-3/2} f(x) e(-ax-bx^{-1}) dx\quad (a,b>0), \] where f(x) is a smooth function compactly supported on (0,\(\infty)\). The core of the argument lies in imitating the aforementioned work of Bombieri-Iwaniec. A new problem arises, which is to count (suitable) integer octuples satisfying certain conditions.

The innovative method of Bombieri-Iwaniec, adapted here to the divisor problem, opens new possibilities in the estimation of exponential sums. It has already incited considerable research: M. N. Huxley and N. Watt [Proc. Lond. Math. Soc. 57, 1-24 (1988; Zbl 0644.10027)] used the technique to obtain new exponent pairs (not obtainable by classical van der Corput theory) and the bound \(\zeta (1/2+it)=O(t^{9/56} \log^ 2t).\) In several other papers in the course of publication they extended the method of Bombieri and Iwaniec. Thus M.N. Huxley [Exponential sums and lattice points, submitted to J. Lond. Math. Soc.] develops a general and powerful method for the estimation of lattice points in planar regions, and in particular improves slightly on the result of the present paper by showing that \[ \Delta (x)=O(x^{7/22}(\log x)^{89/22}). \]

Instead of treating \(\Delta(x)\) by (double) exponential sums via Voronoi’s formula, as was done by several generations of number theorists from E. C. Titchmarsh to G. Kolesnik, the authors adopt a new approach. They start from the elementary formula \[ (1)\quad \Delta (x)=- 2\sum_{n\leq \sqrt{x}}\psi (x/n)+0(1),\quad \psi (t)=t-[t]-1/2 \] and use a variety of techniques. These include approximation by Farey points and ideas from the Weyl and van der Corput treatment of exponential sums, such as Poisson summation. The (technically quite complicated) paper draws on ideas from the pioneering work of E. Bombieri and the first author [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 473-486 (1986; Zbl 0615.10046) and ibid. Ser. 13, 449-472 (1986; Zbl 0615.10047)] in which they proved \(\zeta (1/2+it)=O(t^{9/56+\epsilon}).\) Note that \(7/22<2\cdot (9/56)=0.3214285...,\) so that in a certain sense the methods of the present paper are even more successful, because one generally expects the exponent that one obtains in the divisor problem to be twice the exponent in the problem of the estimation of \(\zeta (1/2+it)\) (see the reviewer’s monograph for a discussion of this), since both problems can be reduced to the estimation of similar exponential sums. However, the present work uses intrinsically (1), whose analogue is not known to hold for \(\zeta (1/2+it),\) and at present it does not seem possible to derive \(\zeta (1/2+it)=O(t^{7/44+\epsilon})\) by the methods of the paper under review.

An important role in the proof is played by “incomplete theta-series” \[ \theta (\alpha,\beta)=\sum_{n}g(n) e(\alpha n+\beta n^ 2)\quad (e(x)=e^{2\pi ix}), \] where \(\alpha\), \(\beta\) are reals and g(n) is a smooth real function, compactly supported. In bounding \(\theta\) (\(\alpha\),\(\beta)\) the authors are led to the evaluation of the integral \[ \int^{\infty}_{0}x^{-3/2} f(x) e(-ax-bx^{-1}) dx\quad (a,b>0), \] where f(x) is a smooth function compactly supported on (0,\(\infty)\). The core of the argument lies in imitating the aforementioned work of Bombieri-Iwaniec. A new problem arises, which is to count (suitable) integer octuples satisfying certain conditions.

The innovative method of Bombieri-Iwaniec, adapted here to the divisor problem, opens new possibilities in the estimation of exponential sums. It has already incited considerable research: M. N. Huxley and N. Watt [Proc. Lond. Math. Soc. 57, 1-24 (1988; Zbl 0644.10027)] used the technique to obtain new exponent pairs (not obtainable by classical van der Corput theory) and the bound \(\zeta (1/2+it)=O(t^{9/56} \log^ 2t).\) In several other papers in the course of publication they extended the method of Bombieri and Iwaniec. Thus M.N. Huxley [Exponential sums and lattice points, submitted to J. Lond. Math. Soc.] develops a general and powerful method for the estimation of lattice points in planar regions, and in particular improves slightly on the result of the present paper by showing that \[ \Delta (x)=O(x^{7/22}(\log x)^{89/22}). \]

Reviewer: A.Ivić

##### MSC:

11N37 | Asymptotic results on arithmetic functions |

11P21 | Lattice points in specified regions |

11L40 | Estimates on character sums |

##### Keywords:

error terms; circle problem; approximation by Farey points; exponential sums; Poisson summation; divisor problem; incomplete theta-series; estimation of exponential sums; exponent pairs; estimation of lattice points; planar regions
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\textit{H. Iwaniec} and \textit{C. J. Mozzochi}, J. Number Theory 29, No. 1, 60--93 (1988; Zbl 0644.10031)

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##### References:

[1] | Bombieri, E; Iwaniec, H, On the order of \(ζ(12 + it)\), Ann. scuola norm. sup. Pisa serie IV, XIII, 449-472, (1986) · Zbl 0615.10047 |

[2] | Chandrasekharan, K, () |

[3] | Gradshteyn, I.S; Ryzhik, I.M, () |

[4] | {\scS. W. Graham and G. Kolesnik}, “Van der Corput Method of Exponential Sums,” Cambridge (to appear). · Zbl 0713.11001 |

[5] | Ivič, A, () |

[6] | Kolesnik, G, On the method of exponent pairs, Acta arith., XLV.2, 115-143, (1985) · Zbl 0571.10036 |

[7] | Titchmarsh, E.C, The theory of the Riemann zeta-function, (1951), Oxford · Zbl 0042.07901 |

[8] | Watson, G.N, A treatise on the theory of Bessel functions, (1944), Cambridge · Zbl 0063.08184 |

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