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.

The mean-square value of the divisor function

Abstract: The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have so thatfor such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.

On primitive roots of unity, divisors of p^2-1, and an extension to mod p^3 of Fermat's Small Theorem

Abstract: Primitive roots of 1 mod p^k (k>2 and odd prime p) are sought, in cyclic units group G_k = A_k B_k mod p^k, coprime to p, of order (p-1)p^{k-1}. 'Core' subgroup A_k has order p-1 independent of k, and p+1 generates 'extension' subgroup B_k of all p^{k-1} residues 1 mod p. Divisors r,t of powerful generator p-1=rs=tu of \pm B_k mod p^k, and of p+1, are investigated as primitive root candidates. Fermat's Small Theorem: x^{p-1} \e 1 mod p for 0 2}. And for prime p: 2^p !=2 and 3^p != 3 (mod p^3). Re: Wieferich primes [4] and FLT case_1. Conj: at least one divisor of p \pm 1 is a semi primitive root of 1 mod p^k. -- (paper withdrawn, re thm2.2)

On zagier's conjecture for l(e,2): a number field example

Abstract: We work out an example, for a CM elliptic curve E defined over a real quadratic field F, of Zagier's conjecture. This relates L(E,2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of E which arises from K-theory. Introduction. Recall that the classical Euler dilogarithm is defined by Li2(z) = ¥ a n=1 z n n 2 |z| < 1

K\"ahler-Ricci flow of cusp singularities on quasi projective varieties

Abstract: Let $\overline{M}$ be a compact complex manifold with smooth Kahler metric $\eta$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hat{\omega}$ be a Carlson-Griffiths type metric on $M$. We study complete solutions to Kahler-Ricci flow on $M$ which are comparable to $\hat{\omega}$, starting from a smooth initial metric $\omega_0=\eta +i\partial \bar{\partial} \phi_0$ where $\phi_0\in C^{\infty}(M)$. When $\omega_0\geq c \hat{\omega}$ on $M$ for some $c>0$ and $\phi_0$ has zero Lelong number, we construct a smooth solution $\omega(t)$ to Kahler-Ricci flow on $M\times [0, T_{[\omega_0 ]})$ where $T_{[\omega_0 ]}:= \sup \{ T: [\eta] +T (c_1(K_{\overline{M}}) + c_1(\mathcal{O}_D))\in \mathcal{K}_M \}$ so that $\omega(t)\geq (\frac{1}{n} - \frac{4\hat{K}t}{c} )\hat{\omega}$ for all $t\leq \frac{c}{4n\hat{K}}$ where $\hat{K}$ is a non-negative upper bound on the bisectional curvatures of $\hat{\omega}$ (see Theorem 1.2). In particular, we do not assume $\omega_0$ has bounded curvature. If $\omega_0$ has bounded curvature and is asymptotic to $\hat{\omega}$ in an appropriate sense, we construct a complete bounded curvature solution on $M\times [0, T_{[\omega_0 ]})$ (see Theorem 1.3). These generalize some of the results of Lott-Zhang in [15]. On the other hand if we only assume $\omega_0\geq c \eta$ on $M$ for some $c>0$ and $\phi_0$ is bounded on $M$, we construct a smooth solution to Kahler-Ricci on $M\times [0, T_{[\omega_0 ]})$ which is equivalent to $\hat{\omega}$ for all positive times. This includes as a special case when $\omega_0$ is smooth on $\overline{M}$ in which case the solution becomes instantaneously complete on $M$ under Kahler-Ricci flow (see Theorem 1.1).

Dynamical invariants and intersection theory on the flex and gothic loci

Abstract: The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new ${\rm SL}_2(\mathbb{R})$-invariant varieties in Teichmuller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-Moller-Sauvaget about computing sums of Lyapunov exponents for ${\rm SL}_2(\mathbb{R})$-invariant varieties via intersection theory.