My main research interests lie within Analytic Number Theory, and particularly in those ideas and results developed in the study of the distribution of primes in small intervals. The methods involved in these now classical results depend on the distribution of zeros of the Riemann zeta-function and Dirichlet L -series, with special regard to the possible exceptional zeros of the latter. In extending these results on rational primes I first looked at Gaussian primes in small regions such as discs D(z,r), centre z and radius r a small power of |z| . For prime ideals in general number fields I then followed Hecke to define similar small regions using Groessencharaktere. The study of problems in such generality requires a detailed study of Hecke L -functions with Groessencharaktere which thus take the place of the Riemann and Dirichlet L -functions above.
So in [3] we give a zero-free region for Hecke L -functions comparable to the classical Vinogradov zero-free region for the Riemann zeta-function. This is fundamental in the following work [5]-[11].
In [7] I followed a method of Selberg to show that, on a form of the Riemann Hypothesis, given a monotonically increasing function F such that F(x)logx→∞, almost all discs in the complex plane, distance r from the origin, and radius F(r)logr, contain a Gaussian prime. The papers [2], [4] and [6] represent an unconditional approach to this problem. The motivation for these questions is the famous (and almost certainly false?) conjecture that one can walk from the origin of the complex plane to infinity, stepping only on Gaussian Primes, and with steps of bounded length.
In [8], [9] and [10] I greatly extended the applicability of a method of Huxley and Hooley in which a clever contour of integration is chosen when trying to estimate integrals of zeta-functions. Amongst applications I gave estimates for the number of Gaussian integers in small discs with a fixed number of prime divisors, a small interval result for the integers at which the Ramanujan tau function lies within a given congruence class, an Erdös-Kac result for ideals in small regions and a very explicit estimation of the number of irreducible integers in small regions.
Also of interest to me are the ideas of sieve theory, especially the Rosser-Iwaniec sieve, which occur in [1], [5] and [6]. In my latest work [11] I and my research student succeeded in proving a Bombieri-Vinogradov theorem for prime ideals in small regions. Amongst many applications this can be used to estimate error terms arising from the application of sieves.
[1] Coleman, M.D. On the equation b1p – b2 P2 = b3, J. reine angew. Math. 403 (1990), 1-66.
[2] Coleman, M.D. The distribution at which binary quadratic forms are prime, Proc. London Math. Soc . (3) 61 (1990), 433-456.
[3] Coleman, M.D. A zero-free region for the Hecke L -functions, Mathematika , 37 (1990), 287-304.
[4] Coleman, M.D. The distribution of points at which norm-forms are prime, J. Number Theory , 41 (1992) 359-378.
[5] Coleman, M.D. The Rosser-Iwaniec sieve in number fields, with an application, Acta Arith. 65 (1993), 53-83.
[6] Coleman, M.D. Relative norms of prime ideals in small regions, Mathematika 43 (1996), 40-62.
[7] Coleman, M.D. The Normal Density of prime ideals in small regions, Monatsh. Math . 125 (1998), 111-126.
[8] Coleman, M.D. The Hooley-Huxley contour method in number fields, I: Arithmetic functions, J. Number Theory 74 (1999), 250-277.
[9] Coleman, M.D. The Hooley-Huxley contour method in number fields II: Factorisation and divisibility, Mathematika . 49 (2002), 201-225
[10] Coleman, M.D. The Hooley-Huxley contour method in number fields III: Frobenian functions, Journal de Theorie des Nombres de Bordeaux , 13 , (2001), 65-74.
[11] Coleman, M.D., Swallow, A.J. Localised Bombieri-Vinogradov theorems in imaginary quadratic fields. 29 pages, Acta Arithmetica . 120 (2005), 349-377.