David Rodney ("Roger") Heath-Brown, F.R.S., is a British mathematician, working in the field of analytic number theory. The Royal Society of London for the Improvement of Natural Knowledge, known simply as The Royal Society, is a Learned society for science that was founded in 1660 In Mathematics, analytic number theory is a branch of Number theory that uses methods from Mathematical analysis to solve number-theoretical problems He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker. Trinity College is a constituent college of the University of Cambridge in Cambridge, England. Alan Baker (born on August 19 1939) is an English Mathematician. He is Professor at the Mathematical Institute of the University of Oxford. The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the
He is known for many striking results. These include an approximate solution to the Artin conjecture on primitive roots, to the effect that out of 3, 5, 7 (or any three similar multiplicatively-independent square-free integers), one at least is a primitive root modulo p, for infinitely many prime numbers p. In Mathematics, the Artin conjecture is a Conjecture on the set of primes p modulo which a given integer a In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 In Modular arithmetic, a branch of Number theory, a primitive root modulo n is any number g with the property that any number Coprime
In collaboration with S. J. Patterson, he in 1978 proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. In Mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3
He has applied Burgess's method on character sums to the ranks of elliptic curves in families. In Mathematics, a character sum is a sum \Sigma \chi(n\ of values of a Dirichlet character &chi Modulo In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O
He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. 
Heath-Brown also showed that Linnik's constant is less than or equal to 5. Linnik's theorem in analytic Number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. 5.