Professor Martin Neil Huxley

Professor of Pure Mathematics, University of Wales, Cardiff

Research interests: prime numbers, Dirichlet series, exponential sums, lattice points and curves, modular forms, eigenvalues of the Laplacian.

Other interests: folk songs, bad verse, history of telephones.

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.