The hexagonal versus the square lattice

We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have `maximal lengths' and goes on to write: "In dimension 2 the conjecture means in particular that the hexagonal lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<... be the positive integers, listed in ascending order, which can be written as h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers, listed in ascending order, which can be written as q_i=x^2+y^2 for integers x and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..." Our proof requires computational prime number theory in combination with methods from a preprint of the first author (to appear in Math. Comp.), arXiv:math.NT/0112100.Comment: 24 pages, 6 figures, 2 table

Average prime-pair counting formula

Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p,\,p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r}\,{\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders \om_{2r}(x)=\pi_{2r(x)-2C_{2r}\,{\rm li}_2(x) that corresponds to Riemann's formula for $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders \om_{2r}(x) which is supported by numerical results

On some conjectural inequalities and their consequences

We discuss some conjectural inequalities involving the sums sig_n(s) := 1^s + 2^s + ... + n^s. Two of our Conjectures assert that both a(s):=sig_(n+1)(s)/sig_n(s) and a(s)*sig_(n+1)(s)/sig_(n+2)(s) are strictly log-convex in s on all of the real axis. We will also present a common generalization of these two Conjectures. Various applications are described, to existing theorems as well as to some other unproven conjecture

The limits of Buchstab's iteration sieve

AbstractAn analysis is presented leading to explicit equations for the limits of the Buchstab iteration sieve. Moreover, the limits are computed for some values of the relevant parameter κ