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The Theory of the Riemann Zeta-Function
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
The density of rational points on curves and surfaces
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
A new form of the circle method, and its application to quadratic forms.
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
The size of Selmer groups for the congruent number problem
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
A mean value estimate for real character sums
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],
Integers Represented as a Sum of Primes and Powers of Two
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
Integer Sets Containing No Arithmetic Progressions
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
The number of primes in a short interval.
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
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