Abstract. A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory. 1. Background For hundreds of years, mathematicians have made conjectures about patterns in the primes: one of the simplest to state is that the primes contain arbitrarily long arithmetic progressions. It is not clear exactly when this conjecture was first formalized, but as early as 1770 Lagrange and Waring studied the problem of how large the common difference of an arithmetic progression of k primes must be. A

We develop the Turán sieve and a ‘simple sieve ’ in the context of bipartite graphs and apply them to various problems in combinatorics. More precisely, we provide applications in the cases of characters of abelian groups, vertex-colourings of graphs, Latin squares, connected graphs, and generators of groups. In addition, we give a spectral interpretation of the Turán sieve.

We show how to control the error term in Mertens ’ formula and related theorems in the context of additive arithmetical semigroups and carry over an old related result of Meissel.

this paper with some simple observations connected to Lambert series and allied matters.

Under the assumption of the well-known heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptography are also discussed. 1

It is argued, that the sums of reciprocals of all consecutive primes separated by gaps of the length d for d 6 are equal to 4c 2 d Q pjd;p?2 p\Gamma1 p\Gamma2 , where c 2 = Q p?2 i 1 \Gamma 1 (p\Gamma1) 2 j . Besides some heuristic arguments leading to this formula, there is also comparison with the results of computer investigation up to 2 42 ß 4:4 \Theta 10 12 . These "experimental" data provides supports for the conjectured formulae. It is also shown, how the guessed formula reproduces the well known fact, that the sum of reciprocals of all primes p ! x grows like ln(ln(x)). 2 Marek Wolf 1. In 1919 Brun [1] has shown that the sum of the reciprocals of all twin primes is finite: B 2 = ` 1 3 + 1 5 ' + ` 1 5 + 1 7 ' + ` 1 11 + 1 13 ' + : : : ! 1: (1) Sometimes 5 is included only once, but here I will adopt the above convention. The analytical formula for B 2 is unknown 1 and the sum (1) is called the Brun constant [2]. The numerical estimations give ...

An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count non-trivial zeros of ζ(s). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.

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We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f(p) where p is a prime and f(x) a polynomial with integer coefficients.