This article exhibits a connection between two well-known problems of number theory: one being the “infinitude of prime values of irreducible polynomials with integer coefficients” and the other being the “non-divisibility of class numbers of imaginary quadratic fields.” Conditionally, we prove Buniakowsky conjecture for an infinite family of polynomials. Some consequences are also discussed.

In this paper, we find a combinatorial identity which is related to the multi-dimensional integral γ m in the study of divisor functions (see (1)). As an application, we determine the finite dual of the group algebra of infinite dihedral group.

Let ε > 0 . We show that every large integer n may be written in the form n = p + a b , where a , b ⩽ n 1 2 − δ for a positive absolute constant δ, and a b ⩽ n 0.55 + ε . This sharpens a result of Heath-Brown (Math. Proc. Cambridge Philos. Soc. 89 (1981), 29–33). The improvement depends on a lower bound version of Bombieri's theorem in short intervals. In establishing such a result, we shall need to “intersect” two lower bound prime-detecting sieves, and we give a more general discussion on this point which may have further applications.

We investigate the relation between the maximum cardinality N of the level sets of a Lipschitz quotient mapping of the plane and the ratio between its Lipschitz and co-Lipschitz constants, with respect to the polygonal norms, and establish that bounds of 1/N previously shown to be sharp for Euclidean norm, stay sharp for polygonal n-norms if and only if n is not divisible by 4.

As a natural analog of Urysohn's inequality in Euclidean space, Gao, Hug, and Schneider showed in 2003 that in spherical or hyperbolic space, the total measure of totally geodesic hypersurfaces meeting a given convex body K is minimized when K is a geodesic ball. We present a random extension of this result by taking K to be the convex hull of finitely many points drawn according to a probability distribution and by showing that the minimum is attained for uniform distributions on geodesic balls. As a corollary, we obtain a randomized Blaschke--Santalo inequality on the sphere.

A conjecture of Leader, Russell and Walters in Euclidean Ramsey theory says that a finite set is Ramsey if and only if it is congruent to a subset of a set whose symmetry group acts transitively. As they have shown the ``if" direction of their conjecture follows if all finite groups have a Hales--Jewett type property. In this paper, we show that this property is satisfied in the case of finite solvable groups. Our result can be used to recover the work of Kříž in Euclidean Ramsey theory.

Thickenings of a metric space capture local geometric properties of the\nspace. Here we exhibit applications of lower bounding the topology of\nthickenings of the circle and more generally the sphere. We explain\ninterconnections with the geometry of circle actions on Euclidean space, the\nstructure of zeros of trigonometric polynomials, and theorems of Borsuk-Ulam\ntype. We use the combinatorial and geometric structure of the convex hull of\norbits of circle actions on Euclidean space to give geometric proofs of the\nhomotopy type of metric thickenings of the circle.\n Homotopical connectivity bounds of thickenings of the sphere allow us to\nprove that a weighted average of function values of odd maps $S^n \\to\n\\mathbb{R}^{n+2}$ on a small diameter set is zero. We prove an additional\ngeneralization of the Borsuk-Ulam theorem for odd maps $S^{2n-1} \\to\n\\mathbb{R}^{2kn+2n-1}$. We prove such results for odd maps from the circle to\nany Euclidean space with optimal quantitative bounds. This in turn implies that\nany raked homogeneous trigonometric polynomial has a zero on a subset of the\ncircle of a specific diameter; these results are optimal.\n

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. As well we establish some upper and lower bounds for the considered integrals, generalizing in particular those obtained by Santal\'o for Masotti's integral.