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.

On the Density of Ranges of Generalized Divisor Functions

Abstract: The range of the divisor function $\sigma_{-1}$ is dense in the interval $[1,\infty)$. However, the range of the function $\sigma_{-2}$ is not dense in the interval $\displaystyle{\left[1,\frac{\pi^2}{6}\right)}$. We begin by generalizing the divisor functions to a class of functions $\sigma_{t}$ for all real $t$. We then define a constant $\eta\approx 1.8877909$ and show that if $r\in(1,\infty)$, then the range of the function $\sigma_{-r}$ is dense in the interval $[1,\zeta(r))$ if and only if $r\leq\eta$. We end with an open problem.

Divisors on varieties over valuation domains

Abstract: LetX be a separated integral normal scheme of finite type over the valuation ringO υ. It is shown that the setD coh(X) of coherent fractionaryO X -ideals $$\mathcal{J} \subseteq \underline {K\left( \mathcal{X} \right)} $$ satisfying the relation $${\mathcal{J}} = \hat {\mathcal{J}} : = ({\mathcal{O}}_{\mathcal{X}} :({\mathcal{O}}_{\mathcal{X}} :{\mathcal{J}}))$$ —the so-called divisorialO X -ideals—forms a group with the composition law $$\left( {\mathcal{I}\mathcal{J}} \right) \mapsto \widehat{\mathcal{I}\mathcal{J}}$$ . This group posesses a natural embedding $$div : \mathcal{D}_{coh} \left( \mathcal{X} \right) \to Div\left( \mathcal{X} \right) \oplus \prod\limits_{v \in V} {v\left( {K\left( \mathcal{X} \right)} \right)} $$ , where Div(X) denotes the group of Weil divisors of the generic fibreX of $$\mathcal{X}\left| {Spec\left( {\mathcal{O}_ u } \right)} \right.$$ , and V is a set of valuations ofK X determined by a subset of the generic points of the fibres $$\mathcal{X} \times _{\mathcal{O}_ u } \mathcal{K}\left( \mathcal{P} \right),\mathcal{P} \in Spec\left( {\mathcal{O}_ u } \right)\backslash 0$$ . The image Div(X) of div is proved to satisfy $$Div(\mathcal{X}) = Div(\mathcal{X}) \oplus Ver(\mathcal{X})$$ with a subgroup $$Ver(\mathcal{X}) \subseteq \prod {_{v \in V} } v\left( {K\left( \mathcal{X} \right)} \right)$$ . The structure of Ver(X) is determined provided thatX satisfies additional conditions—for example, ifX is projective over Spec(O v ). These facts are deduced from general results on the semigroupD coh(X) of coherent divisorialO X -ideals on an integral schemeX: A criterion forD coh(X) to be a group based on the notion of so-called Pruferv-multiplication rings, and a valuation theoretic description of this group using valuations of K(X) naturally associated toX. The considerations leading to these results show thatD coh(X) can be understood as an ideal theoretic generalization of the group of Weil divisors on a normal noetherian scheme. Following this idea a criterion forD coh(X),X a separated integral normalO v -scheme of finite type, to be equal to the group of Cartier divisors onX is given. The criterion is obtained by showing that for any pointx on such a scheme the local generalized Weil divisor groupsD coh(Spec(O,x)) exist and by analyzing the structure of these groups.

On the Dirichlet divisor problem in short intervals

Abstract: We present several new results involving $\Delta(x+U)-\Delta(x)$, where $U = o(x)$ and $$ \Delta(x):=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.

On the Exponential Divisor Function

Abstract: We investigate the exponential divisor function and establish several asymptotic formulas involving this function.

The mean value theorem of the divisor problem for short intervals

Abstract: For $-1\leq a \leq 0$ let ${\it\Delta} _{a}(x;r)$ denote the error term in the asymptotic formula for $\sum \limits _{n \leq x} {\it\sigma} _{a}(n)e(rn)$ . We study the mean square of the function $|{\it\Delta} _{a}(x+U;r)-{\it\Delta} _{a}(x;r)|$ for short intervals.