Let Q(x; y) be a primitive positive de nite quadratic form with integer coecients. Then, for all (s; t) 2 R 2 there exist (m; n) 2 Z 2 such that Q(m;n) is prime and Q(m s; n t) Q(s; t) 0:53 : This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.

. Given a real vector ff=(ff1 ; : : : ; ff d ) and a real number " ? 0 a good Diophantine approximation to ff is a number Q such that kQff mod Zk1 ", where k \Delta k1 denotes the `1-norm kxk1 := max 1id jx i j for x = (x1 ; : : : ; xd ). Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector ff 2 Q d , a rational number " ? 0 and a number N 2 N+ , decide whether there exists a number Q with 1 Q N and kQff mod Zk1 ". We prove that, unless NP ` DTIME(n poly(log n) ), there exists no polynomial -time algorithm which computes on inputs ff 2 Q d and N 2 N+ a number Q with 1 Q 2 log 0:5\Gammafl d N and kQ ff mod Zk1 2 log 0:5\Gammafl d min 1qN jjqff mod Zk1 ; where fl is an arbitrary small positive constant. To put it in other words, it is almost NP--hard to approximate a minimum good Diophantine approximation to ff in polynomial-time within a factor 2 log 0:5\Gammafl d for an arbitrary small positive const...

We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)-#(x-y) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, and also on the strength of density bounds for the zeros themselves. We also study implications in the opposite direction: assuming that an asymptotic formula like the above is valid for almost all x in a given range of values for y, we find zero-free regions or density bounds.

Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum

These are notes of a series of lectures on sieves, presented during the Special

ABSTRACT. Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h, k), which counts the maximum possible number of essential subsets of size k, in a basis ( of order h. For a fixed k and with h going to infinity, we show that k 1/(k+1) E(h, k) = Θk [h / log h] ). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known ‘postage stamp problem’ in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show log k that E(h, k) ∼ (h − 1) log log k. 1. ESSENTIAL SUBSETS OF INTEGER BASES Let S be a countable abelian semigroup, written additively, h be a positive integer and A ⊆ S. The h-fold sumset hA consists of all s ∈ S which can be expressed as