Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)Jurkat-Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec's lower bound on Jacobsthal's problem, as well as minor improvements to the Brun-Titchmarsh Theorem. We also deduce an improved (though conditional) lower bound on the longest gaps between primes, and rework Cram\'er's heuristic in this situation to show that we would expect gaps around \( x\) that are significantly larger than \( (\log x)^2.\)

Let \( A \subset \mathbb{Z}^d\) be a finite set. It is known that \( NA\) has a particular size (\( \vert NA\vert = P_A(N)\) for some \( P_A(X) \in \mathbb{Q}[X]\) ) and structure (all of the lattice points in a cone other than certain exceptional sets), once \( N\) is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary \( A\) . Such explicit results were only previously known in the special cases when \( d=1\) , when the convex hull of \( A\) is a simplex or when \( \vert A\vert = d+2\) , results which we improve.

We compare appreciation of mathematics with appreciation of impressionist art

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Polya-Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess' estimate for short character sums, and upper bounds for \(L(1,\chi)\) and \( L(1+it,\chi)\)) are more-or-less ``equivalent''. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

For a given positive integer \( k\), we prove that there are at least \( x^{1/2-o(1)}\) integers \(d\leq x\) such that the real quadratic fields \(\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})\) have class numbers essentially as large as possible, that is \(\gg_k \sqrt{d} \frac{\log\log d}{\log d}\)