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.