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The Theory of the Riemann Zeta-Function

- E. C. Titchmarsh, D. R. Heath-Brown
- Mathematics
- 5 February 1987

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects… Expand

The density of rational points on curves and surfaces

- D. R. Heath-Brown, J.-L. Colliot-Th'elene
- Mathematics
- 1 March 2002

Let $C$ be an irreducible projective curve of degree $d$ in $\mathbb{P}^3$, defined over $\overline{\mathbb{Q}}$. It is shown that $C$ has $O_{\varepsilon,d}(B^{2/d+\varepsilon})$ rational points of… Expand

Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression

- D. R. Heath-Brown
- Mathematics
- 1 March 1992

A new form of the circle method, and its application to quadratic forms.

- D. R. Heath-Brown
- Mathematics
- 1996

If the coefficients r(n) satisfy suitable arithmetic conditions the behaviour of F (α) will be determined by an appropriate rational approximation a/q to α, with small values of q usually producing… Expand

The size of Selmer groups for the congruent number problem

- D. R. Heath-Brown
- Mathematics
- 1 December 1993

where Np is the number of solutions of the congruence y ≡ x −D2x (mod p). Then LD(s) has an analytic continuation as an entire function on the complex plane. The conjecture of Birch and… Expand

A mean value estimate for real character sums

- D. R. Heath-Brown
- Mathematics
- 1995

where Σ∗ indicates summation over primitive characters only. These last two bounds follow respectively from Theorem 6.2 of Montgomery [6] and from the large sieve in the form due to Gallagher [2],… Expand

Integers Represented as a Sum of Primes and Powers of Two

- D. R. Heath-Brown, Jan-Christoph Puchta
- Mathematics
- 30 January 2002

It is shown that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. Under the Generalized Rieman Hypothesis one can replace 13 by 7. Unlike previous work on this… Expand

Integer Sets Containing No Arithmetic Progressions

- D. R. Heath-Brown
- Mathematics
- 1 June 1987

lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of… Expand

The number of primes in a short interval.

- D. R. Heath-Brown
- Mathematics
- 1988

which estimates the number of primes in the interval (x —y, x]. According to the Prime Number Theorem, the above estimate holds uniformly for cx^y^x, if c is any positive constant. Much work has been… Expand

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