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.

Convolution sums of some functions on divisors

Abstract: One of the main goals in this paper is to establish convolution sums of functions for the divisor sums $\widetilde{\sigma}_s(n)=\sum_{d|n}(-1)^{d-1}d^s$ and $\widehat{\sigma}_s(n)=\sum_{d|n}(-1)^{\frac{n}{d}-1}d^s$, for certain $s$, which were first defined by Glaisher. We first introduce three functions $\mathcal{P}(q)$, $\mathcal{E}(q)$, and $\mathcal{Q}(q)$ related to $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, respectively, and then we evaluate them in terms of two parameters $x$ and $z$ in Ramanujan's theory of elliptic functions. Using these formulas, we derive some identities from which we can deduce convolution sum identities. We discuss some formulae for determining $r_s(n)$ and $\delta_s(n)$, $s=4,$ $8$, in terms of $\widetilde{\sigma}(n)$, $\widehat{\sigma}(n)$, and $\widetilde{\sigma}_3(n)$, where $r_s(n)$ denotes the number of representations of $n$ as a sum of $s$ squares and $\delta_s(n)$ denotes the number of representations of $n$ as a sum of $s$ triangular numbers. Finally, we find some partition congruences by using the notion of colored partitions.

Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors

Abstract: Let $X$ be a non-singular compact Kahler manifold, endowed with an effective divisor $D= \sum (1-\beta_k) Y_k$ having simple normal crossing support, and satisfying $\beta_k \in (0,1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X, D)$ are the so-called metrics with conic singularities. In this article, we complete our earlier work \cite{CGP} concerning the Monge-Ampere equations on $(X, D)$ by establishing Laplacian and ${\mathscr C}^{2,\alpha, \beta}$ estimates for the solution of this equations regardless to the size of the coefficients $0<\beta_k< 1$. In particular, we obtain a general theorem concerning the existence and regularity of Kahler-Einstein metrics with conic singularities along a normal crossing divisor.

Spaces of surface group representations

Abstract: Let $$\Gamma _g$$ denote the fundamental group of a closed surface of genus $$g \ge 2$$ . We show that every geometric representation of $$\Gamma _g$$ into the group of orientation-preserving homeomorphisms of the circle is rigid, meaning that its deformations form a single semi-conjugacy class. As a consequence, we give a new lower bound on the number of topological components of the space of representations of $$\Gamma _g$$ into $${{\mathrm{Homeo}}}_+(S^1)$$ . Precisely, for each nontrivial divisor $$k$$ of $$2g-2$$ , there are at least $$|k|^{2g} + 1$$ components containing representations with Euler number $$\frac{2g-2}{k}$$ . Our methods apply to representations of surface groups into finite covers of $${{\mathrm{PSL}}}(2,\mathbb {R})$$ and into $${{\mathrm{Diff}}}_+(S^1)$$ as well, in which case we recover theorems of W. Goldman and J. Bowden. The key technique is an investigation of stability phenomena for rotation numbers of products of circle homeomorphisms using techniques of Calegari–Walker. This is a new approach to studying deformation classes of group actions on the circle, and may be of independent interest.