Professor Martin Neil Huxley
Professor of Pure Mathematics, University of Wales, Cardiff
Email address: Huxley@cf.ac.uk
Research interests: prime numbers, Dirichlet series, exponential
sums, lattice points and curves, modular forms, eigenvalues of the
Other interests: folk songs, bad verse,
history of telephones.
- 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
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
- 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
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
- 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
in Number Theory in Memory of Kustaa Inkeri, De Gruyter, Berlin
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
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).
- 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
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
Originally part of the long preprint version of 72.