Abstract. Let ψ(x) be a polynomial with rational coefficients. Suppose that ψ has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x, y ∈ A and a prime p such that x− y = ψ(p− 1). Furthermore, if P is a set of primes with the positive relative upper density, then there exist x, y ∈ P and a prime p such...

Let [Formula: see text] be a real and non-principal Dirichlet character, its text]-function let generic prime number. We prove the following result: If for some partial sums change sign only finite number of integers text], then there exists such that has no zeros in half plane text].

Let R be a polynomial ring in finitely many variables over the integers, and fix an ideal a of R. We prove that for all but finitely prime integers p, the Bockstein homomorphisms on local cohomology, H a (R/pR) −→ H k+1 a (R/pR), are zero. This provides strong evidence for Lyubeznik’s conjecture which states that the modules H a (R) have a finite number of associated prime ideals.

Let λ1, . . . , λs be non-zero with λ1/λ2 irrational and let S be the set of values attained by the form λ1x 3 1 + · · ·+ λsxs when x1 has at most 6 prime divisors and the remaining variables are prime. In the case s = 4, we establish that most real numbers are “close” to an element of S. We then prove that if s = 8, S is dense on the real line.

Let k be a field, possibly of finite characteristic, V a finite dimensional vector space over k of dimension r, and W ⊂ Autk(V ) a finite subgroup. There is a natural action of W on the k-dual V # of V , as well as on the symmetric algebra S = S(V ). The algebra S is isomorphic to a polynomial algebra over k on r generators, and we are interested in the question of when the invariant algebra R ...

We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.

In this paper we study elementary extensions of differential fields in prime characteristic. In particular, we show that, in contrast to Liouville’s result in characteristic zero, all elements of an elementary extension admit an antiderivative in some logarithmic extension.

We show that the fundamental group of a prime alternating surfacearc complement is δ-hyperbolic in case the genus of the surface is greater than zero. AMS Subject classification: 57M25, 57M50, 57M05