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.

Exact Asymptotics in log log Laws for Random Fields

Abstract: Let $$\{ X,X_k ,k \in {\mathbb{N}}^r \}$$ be i.i.d. random variables, and set S n =∑ k ≤ n X k . We exhibit a method able to provide exact loglog rates. The typical result is that $${\mathop {\lim }\limits_{\varepsilon \searrow \sigma \sqrt {2r}} } \sqrt {\varepsilon ^2 - 2r\sigma ^2 } \sum\limits_n {\frac{1}{{|\,n\,|}}P(|S_n \geqslant \varepsilon \sqrt {|\,n\,|\log \log |\,n\,|} ) = \frac{{\sigma \sqrt {2r} }}{{r!}},}$$ whenever EX=0,EX 2=σ2 and E[X 2(log+ | X |) r-1] < ∞. To get this and other related precise asymptotics, we derive some general estimates concerning the Dirichlet divisor problem, of interest in their own right.

Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem

Abstract: Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also placewhere γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesaro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

Monodromy of a family of hypersurfaces

Abstract: Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta$ be a positive integer such that $\mathcal I_{Z,Y}(\delta)$ is generated by global sections. Fix an integer $d\geq \delta +1$, and assume the general divisor $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ the quotient of $H^m(X;\mathbb Q)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ for the family of smooth divisors $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is irreducible.

Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field

Abstract: Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, \Delta)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q}$-divisor $\Delta' \sim_{\mathbb Q} \Delta+D$ such that $(X, \Delta')$ is log canonical if there exists a log resolution of $(X, \Delta)$.

Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem

Abstract: We calculate the triple correlations for the truncated divisor sum $\lambda_{R}(n)$. The $\lambda_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $\Lambda(n)$ does or is conjectured to do. We also calculate the mixed (with a factor of $\Lambda(n)$) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation $\Lambda_{R}(n)$. However, when $\lambda_{R}(n)$ is used the error in the singular series approximation is often much smaller than what $\Lambda_{R}(n)$ allows. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain an $\Omega_{\pm}$-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to $\Omega$-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on $\lambda_{R}(n)$'s and $\Lambda_{R}(n)$'s can be employed in diverse problems concerning primes.