Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Subconvexity for the Riemann zeta function and the divisor problem

Abstract: A simple proof of the classical subconvexity bound $\zeta(1/2+it) \ll_\epsilon t^{1/6+\epsilon}$ for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor problem and the mean square of $|\zeta(1/2+it)|$ are analysed.

Isomonodromic deformations of logarithmic connections and stability

Abstract: Let \(X_0\) be a compact connected Riemann surface of genus g with \(D_0 \subset X_0\) an ordered subset of cardinality n, and let \(E_G\) be a holomorphic principal G-bundle on \(X_0\), where G is a reductive affine algebraic group defined over \(\mathbb C\), that is equipped with a logarithmic connection \( abla _0\) with polar divisor \(D_0\). Let \((\mathcal {E}_G , abla )\) be the universal isomonodromic deformation of \((E_G , abla _0)\) over the universal Teichmuller curve \((\mathcal {X}, \mathcal {D})\,{\longrightarrow }\, \text {Teich}_{g,n}\), where \(\text {Teich}_{g,n}\) is the Teichmuller space for genus g Riemann surfaces with n–marked points. We prove the following (see Sect. 5): (1) Assume that \(g \ge 2\) and \(n= 0\). Then there is a closed complex analytic subset \(\mathcal {Y} \subset \text {Teich}_{g,n}\), of codimension at least g, such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is semistable, where \({\mathcal X}_t\) is the compact Riemann surface over t. (2) Assume that \(g\ge 1\), and if \(g= 1\), then \(n > 0\). Also, assume that the monodromy representation for \( abla _0\) does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \(\mathcal {Y}' \subset \text {Teich}_{g,n}\), of codimension at least g, such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is semistable. (3) Assume that \(g\ge 2\). Assume that the monodromy representation for \( abla _0\) does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \(\mathcal {Y}'' \subset \text {Teich}_{g,n}\), of codimension at least \(g-1\), such that for any \(t\in \text {Teich}_{g,n} {\setminus } \mathcal {Y}'\), the principal G-bundle \(\mathcal {E}_G\vert _{{\mathcal X}_t}\) is stable. In [12], the second-named author proved the above results for \(G= \text {GL}(2,{\mathbb C})\).

On the greatest common divisor of polynomials which depend on a parameter

Abstract: The following computer algebra problem is considered : how to compute the gcd of the polynomials U(X, a) and V(Z, a) for various values of the parameter a? This problem appears, for example, in solving systems of algebraic equations by the elimination methods ~aer53], in computing the logarithmic part of the integral of a rational function [Trag76, Laz&Rio90], in solving difference and differential equations [Abr89], in summing rational functions [Abr71, Abr75, Gosp78], etc... A fast algorithm to solve this problem is described, and some applications of this algorithm are discussed.

Lower Bounds for Integer Greatest Common Divisor Computations (Extended Summary)

Abstract: We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.

Primitive divisors on twists of the fermat cubic

Abstract: We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u 3 + v 3 = m, with m cube-free, all the terms beyond the first have a primitive divisor.