### Recent Publications

- 50
__A mean value theorem for exponential sums__,

*Journal of the Australian Math. Soc.*A59 (1995), 304-307.

Replace f(m) by f(m+x), average x from 0 to 1. - 53 with G. Kolesnik,
__Exponential sums with a large second derivative__, (revised version)

in Number Theory in Memory of Kustaa Inkeri, de Gruyter, Berlin (2001), 131-144.

A different construction of resonance curves, useful when (log M)/(log T) is near 4/7. Revised version, using paper 69. - 56
__The mean lattice point discrepancy__,

*Proceedings of the Edinburgh Math. Soc.*38 (1995), 523-531.

The discrepancy has no memory of its previous values more than a minute ago. - 57 Area, Lattice Points, and Exponential Sums,

*London Mathematical Society Monographs*13 (1966), 506pp.

Finding the area by counting squares, and many related topics. - 58 with Trifonov,
__The square-full numbers in an interval__,

*Math. Proc. Cambridge Philosophical Society*119 (1996), 201-208.

A sharper bound for the number of integer points close to a curve is used to show that a shortish interval contains the expected number of square-full numbers. - 59 with Sargos,

__Points entiers au voisinage d’une courbe plane de classe C^n__,

*Acta Arithmetica*69 (1995), 359-366.

Integer points close to a curve are either spread out (minor arcs) or on algebraic curves (major arcs). - 60 with Hall and Wilson,

__A three variable identity connected with Dedekind sums__,

*Periodica Math. Hungarica*39 (1995), 189-203.

Cases when the symmetric sum of three Franel integrals simplifies. - 61
__Moments of differences between square-free numbers__,

in Sieve Methods, Exponential Sums, and their Applications

in Number Theory, London Math. Soc. Lecture Notes 237, Cambridge (1997), 187-204.

Uses elementary methods to count triples of numbers divisible by the square of a large prime. - 62 with N. Watt,
__The number of ideals in a quadratic field II__,

*Israel J. Math. 120 (2001), 125-153.*

Builds a quadratic character into the Iwaniec-Mozzochi method for counting lattice points. - 63
__The integer points close to a curve II__,

in Analytic Number Theory, Birkhäuser (1996), 487-516.

A constructive lower bound for the number of integer points close to a curve on major arcs. - 65 with Nowak,
__Primitive lattice points in convex planar domains__,

*Acta Arithmetica*56 (1996), 271-283.

Counting the number of lattice points visible from the origin.

Needs the Riemann Hypothesis, though. - 66 with Watt,
__Congruence families of exponential sums__,

in Analytic Number Theory, London Math. Soc. Lecture Notes 247, Cambridge (1997), 127-138.

Gets a bound for the Dirichlet L-function with exponents 89/570 in t-aspect, 2/5 in q-aspect. - 67
__The shear difficulty in lattice point problems__,

in Proceedings of the Conference on Analytic and Elementary Number Theory, Vienna University (1997), 92-111.

A flattish curve with a perturbing linear term (or a skew lattice), so the gradient is nearly constant. The continued fraction for the gradient enters the estimates. - 68 with Watt,
__Hybrid bounds for Dirichlet’s L-function__,

*Math. Proc. Camb.Philos. Soc.*129 (2000), 385-415.

More complicated bounds for the Dirichlet L-functions, considering both t-aspect and q-aspect. - 69
__The integer points close to a curve III__,

in Number Theory in Progress, de Gruyter, Berlin (1999), 911-940.

Swinnerton-Dyer’s upper bound extended to arcs with large gradient. - 70
__The rational points close to a curve II__,

*Acta Arithmetica*93 (2000), 201-219.

The values of x and y are rational numbers with different denominators, and the major arcs are linear fractional curves. - 74
__Integer points in plane regions and exponential sums__,

in Number Theory, Birkhäuser, Basel (2000), 157-166.

A survey of recent work and how the two types of problem are connected. - 75 with A. A. Zhigljavsky,
__On the distribution of Farey fractions and hyperbolic lattice points__,

*Periodica Math. Hung*, 42 (2001), 191-198.

The distribution of consecutive pairs of Farey fractions leads to an elementary estimate for the number of images of a given point in a large circle under the action of the modular group in hyperbolic space. - 76 with G. R. H. Greaves,
__One-sided sieving density hypothesis in Selberg’s sieve__,

in Number Theory in Memory of Kustaa Inkeri, De Gruyter, Berlin (2001), 105-114.

The small sieve upper bound is usually stated in terms of the average number of residue classes removed for each prime. Assuming only a lower bound on average leads to the same accuracy.

See Greaves’s Web page for correction. - 78
__Integer points, exponential sums and the Riemann zeta function__

in Number Theory for the Millennium, A. K. Peters (2002) vol II, 275-290.

A survey of recent results, with a sketch proof of the exponent 137/432 for mean squares of exponential sums (Titchmarsh’s E(T) problem for the zeta function).

### Forthcoming Publications

- 72
__Exponential Sums and the Riemann zeta function V__

Sets up a new Step in which results on integer points close to curves lead to small improvements for van der Corput exponential sums. The latest results by Swinnerton-Dyer’s method lead to exponent 32/205 in the Lindelöf problem. - 73
__Exponential Sums and Lattice Points III__

*Proc. London Math. Soc.*(2003).

A notional iteration from integer points close to curves to the lattice point discrepancy. The latest result by Swinnerton-Dyer’s method leads to exponents 131/208 in the circle problem, 131/416 in the divisor problem. - 79
__A determinant mean value theorem__

A mean value theorem for the determinant of n functions evaluated at n points, used in later papers on points close to curves, which I like better than certain journal editors do. - 80
__The rational points close to a curve III__

*Acta Arithmetica*

Points x = m/n, y = r/q very close to a given curve, with n of size M, q of size Q, and Q allowed to be larger than M. `Very close’ means that there are no major arcs. - 81
__The rational points close to a curve IV__

in Proceedings of the Bonn Semester.

Points x = m/n, y = r/q fairly close to a given curve, with n of size M, q of size Q, and Q allowed to be larger than M. There are major arcs: regions where the curve can be approximated by a rational function of low degree. - 82
__Resonance Curves in the Bombieri-Iwaniec Method__

Estimating an exponential sum with phase function f(x) is like the lattice point problem for the underlying curve y = f'(x). Resonances occur when an affine map that fixes the integer lattice superposes one arc of the underlying curve onto another arc (modulo the integer lattice). For a given affine map, a test for resonances is whether there is an integer point close to a certain plane curve. This `resonance curve’ is properly constructed as the solution to a differential equation, with better approximation properties, and a functorial property under inclusion.

Originally part of the long preprint version of 72.

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