Showing papers on "Divisor published in 2007"

The average value of divisor sums in arithmetic progressions

Abstract: Let α(n) denote the Fourier coefficients of cusp forms or the number of divisors of n. Estimates of the typeare shown, uniformly in q ≤ X. The methods can be extended to other arithmetic functions, for example, the number of representations of n as a sum of two squares or k-free numbers. As an application, sums of the type ∑ n ≤ X α(n) ψ(n) for any q-periodic function ψ can be estimated non-trivially.

On the higher moments of the error term in the divisor problem

Abstract: Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas for the integral of the cube and the fourth power of $\Delta(x)$. The exponents that we obtain in the error terms, namely $\beta = {\sfrac{7}{5}}$ and $\gamma = {\sfrac{23}{12}}$, respectively, are new. They improve on the values $\beta = {\sfrac{47}{28}}, \gamma = {\sfrac{45}{23}}$, due to K.-M. Tsang. A result on integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals is also proved.

Primitive Divisors in Arithmetic Dynamics

Abstract: Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.

Primes generated by recurrence sequences

Abstract: We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.

Fano 3-folds with divisible anticanonical class

Abstract: We show the nonvanishing of H0(X,−KX) for any a Fano 3-fold X for which −KX is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, \({\mathbb{Q}}\) -factorial terminal singularities and −KX = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H0(X,−KX) and the sharp bound (−KX)3≥ 8/165. We find the families that can be realised in codimension up to 4.

Classification of Fano manifolds containing a negative divisor isomorphic to projective space

Abstract: We classify n-dimensional complex Fano manifolds X (n ≥ 3) containing a divisor E isomorphic to \({\mathbb{P}^{n-1}}\) such that deg NE/X is strictly negative. Our main tool is the extremal contraction theory together with numerical arguments on intersection numbers of divisors on X. In the last section, we consider, more generally, Fano manifolds X containing a prime divisor with Picard number one, and show that the Picard number of such X is less than or equal to three.

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.

Toroidal Embeddings and Polyhedral Divisors

Abstract: Given an effective action of an (n−1)-dimensional torus on an n- dimensional normal affine variety, Mumford constructs a toroidal embedding, while Altmann and Hausen give a description in terms of a polyhedral divisor on a curve. We compare the fan of the toroidal embedding with this polyhedral divisor.

Extended Golomb Code for Integer Representation

Abstract: In this paper, we have proposed two methods to represent nonnegative integers based on the principle used in Golomb code (GC). In both methods, the given integer is successively divided with a divisor, the quotient and the remainders are then used to represent the integer. One of our methods is best suited for representing short integers and gives bit length comparable to that of Elias radic code which is best for representing short-range integers. Another of our methods is best suited for representing both short and long integers and gives a bit length comparable to that of Fibonacci code which is best for representing long integers. Application of our methods as a final stage encoder of the Burrows-Wheeler transform compressor shows that our codes give a better compression rate than the Elias, Fibonacci, punctured, and GC codes

Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

Abstract: Let S_{w+2} be the vector space of cusp forms of weight w+2 on the full modular group, and let S_{w+2}^* denote its dual space. Periods of cusp forms can be regarded as elements of S_{w+2}^*. The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span S_{w+2}^*. However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural question: which periods would form a basis of S_{w+2}^*. First we give an answer to this question. Passing to the dual space S_{w+2}, we will determine a new basis for S_{w+2}. The even period polynomials of this basis elements are expressed explicitly by means of Bernoulli polynomials. Next we consider three spaces--S_{w+2}, the space of even Dedekind symbols of weight w with polynomial reciprocity laws, and the space of even period polynomials of degree w. There are natural correspondences among these three spaces. All these spaces are equipped with compatible action of Hecke operators. We will find explicit form of period polynomials and the actions of Hecke operators on the period polynomials. Finally we will obtain explicit formulas for Hecke operators on S_{w+2} in terms of Bernoulli numbers B_k and divisor functions sigma_k(n), which are quite different from the Eichler-Selberg trace formula.

Logarithmic comparison theorem and D-modules: an overview

Abstract: . Let D ⊂ X be a divisor in a complex analytic manifold. A naturalproblem is to determine when the de Rham complex of meromorphic forms onX with poles along D is quasi-isomorphic to its subcomplex of logarithmicforms. In this mostly expository note, we recall the main results about thisproblem. In particular, we point out the relevance of the theory of D-modulesto this topic. Introduction Let X be a complex analytic manifold of dimension n ≥ 2. Given a di-visor D ⊂ X, we denote j the natural inclusion X\D → X. Let Ω •X (⋆D)denote the complex of meromorphic forms on X with poles along D. From theGrothendieck Comparison Theorem [17], the de Rham morphismΩ •X (⋆D) −→ Rj ∗ C X\D is a quasi-isomorphism. In particular, if X = C n , then for each cohomologyclass c ∈ H p (C n \D,C), there exists a differential form w ∈ Ω pX (⋆D) such thatfor any p-cycle σ on C n \D, one has c(σ) =R σ w.It is natural to ask what one can say about the form w. For example, if Dis a complex submanifold then the order of the pole of w can be taken to be1. The question of the order of the pole goes back to P.A. Griffiths [16]. Werecall that a meromorphic form w ∈ Ω

Efficient explicit formulae for genus 2 hyperelliptic curves over prime fields and their implementations

Abstract: We analyze all the cases and propose the corresponding explicit formulae for computing 2D1 + D2 in one step from given divisor classes D1 and D2 on genus 2 hyperelliptic curves defined over prime fields. Compared with naive method, the improved formula can save two field multiplications and one field squaring each time when the arithmetic is performed in the most frequent case. Furthermore, we present a variant which trades one field inversion for fourteen field multiplications and two field squarings by using Montgomery's trick to combine the two inversions. Experimental results show that our algorithms can save up to 13% of the time to perform a scalar multiplication on a general genus 2 hyperelliptic curve over a prime field, when compared with the best known general methods.

Index Calculus Attack for Jacobian of Hyperelliptic Curves of Small Genus Using Two Large Primes

Abstract: This paper introduces a fast algorithm for solving the DLP of Jacobian of hyperelliptic curve of small genus. To solve the DLP, Gaudry first shows that the idea of index calculus is effective, if a subset of the points of the hyperelliptic curve of the base field is taken by the smooth elements of index calculus. In an index calculus theory, a special element (in our case it is the point of hyperelliptic curve), which is not a smooth element, is called a large prime. A divisor, written by the sum of several smooth elements and one large prime, is called an almost smooth divisor. By the use of the almost smooth divisor, Theriault improved this index calculus. In this paper, a divisor, written by the sum of several smooth elements and two large primes, is called a 2-almost smooth divisor. By use of the 2-almost smooth divisor, we are able to give more improvements. The algorithm of this attack consists of the following seven parts: 1) Preparing, 2) Collecting reduced divisors, 3) Making sufficiently large sets of almost smooth divisors, 4) Making sufficiently large sets of smooth divisors, 5) Solving the linear algebra, 6) Finding a relation of collected reduced divisors, and 7) Computing a discreet logarithm. Parts 3) and 4) need complicated eliminations of the large prime, which is the key idea presented within this paper. Before the tasks in these parts are completed, two sub-algorithms for the eliminations of the large prime have been prepared. To explain how this process works, we prove the probability that this algorithm does not work to be negligible, and we present the expected complexity and the expected storage of the attack.

Log minimal model program for the moduli space of stable curves of genus three

Abstract: In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple $\alpha\delta$ of the divisor $\delta$ of singular curves as the boundary divisor, construct the log canonical model for the pair $(\bar{\mathcal M}_3, \alpha\delta)$ using geometric invariant theory as we vary $\alpha$ from one to zero, and give a modular interpretation of each log canonical model and the birational maps between them. By using the modular description, we are able to identify all but one log canonical models with existing compactifications of $M_3$, some new and others classical, while the exception gives a new modular compactification of $M_3$.

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.

Modular unit and cuspidal divisor class groups of X_1(N)

Abstract: In this article, we consider the group $F_1^\infty(N)$ of modular units on $X_1(N)$ that have divisors supported on the cusps lying over $\infty$ of $X_0(N)$, called the $\infty$-cusps. For each positive integer $N$, we will give an explicit basis for the group $F_1^\infty(N)$. This enables us to compute the group structure of the rational torsion subgroup $C_1^\infty(N)$ of the Jacobian $J_1(N)$ of $X_1(N)$ generated by the differences of the $\infty$-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the $p$-primary part of $C_1^\infty(p^n)$ for a regular prime $p$.

A remark on pseudoconvex domains with analytic complements in compact Kähler manifolds

Abstract: For an effective divisor $A$ with support $B$ in a compact Kahler manifold $M$ of dimension $\geq 3$, the following are antinomic. a) $M\backslash B$ has a $C^{\infty}$ plurisubharmonic exhaustion function whose Levi form has pointwise at least 3 positive eigenvalues outside a compact subset of $M\backslash B$. b) $[A]|B$, the normal bundle of $A$, is topologically trivial.

The Number of Rational Curves on K3 Surfaces

Abstract: Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it verifies Yau-Zaslow formula in the non primitive class $\beta$.

Sharp probability estimates for Shor's order-finding algorithm

Abstract: Let N be a (large) positive integer, let b be an integer satisfying 1 < b < N that is relatively prime to N, and let r be the order of b modulo N. Finally, let QC be a quantum computer whose input register has the size specified in Shor's original description of his order-finding algorithm. In this paper, we analyze the probability that a single run of the quantum component of the algorithm yields useful information--a nontrivial divisor of the order sought. We prove that when Shor's algorithm is implemented on QC, then the probability P of obtaining a (nontrivial) divisor of r exceeds.7 whenever N ≥ 211 and r ≥ 40, and we establish that.7736 is an asymptotic lower bound for P. When N is not a power of an odd prime, Gerjuoy has shown that P exceeds 90 percent for N and r sufficiently large. We give easily checked conditions on N and r for this 90 percent threshold to hold, and we establish an asymptotic lower bound for P of 2Si(4π)/π ≈.9499 in this situation. More generally, for any nonnegative integer q, we show that when QC(q) is a quantum computer whose input register has q more qubits than does QC, and Shor's algorithm is run on QC(q), then an asymptotic lower bound on P is 2Si(2q+2π)/π (if N is not a power of an odd prime). Our arguments are elementary and our lower bounds on P are carefully justified.

Method and apparatus for integer division

Abstract: A method, arithmetic divider unit, and system are disclosed for dividing a dividend D ZM . . . Z0 having a most significant bit Z M and a plurality of less significant bits Z M−1 through Z 0 by a divisor R ZN . . . Z0 having a most significant bit Z N and a plurality of less significant bits Z N−1 through Z 0 . The method, arithmetic divisor unit, and system round the divisor to the next significant bit greater than the divisor's most significant bit Z N to produce a first partial divisor R ZN+1 , divide the dividend D ZM . . . Z0 by the first partial divisor R ZN+1 to produce a first partial quotient Q N , calculate one or more additional partial quotients based on one or more divisor bits selected from the plurality of divisor bits Z N−1 through Z 0 , and add the first partial quotient Q N and one or more additional partial quotients to produce an estimated final quotient.

An order result for the exponential divisor function

Abstract: The integer $d=\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \mid a_i$ for every $i\in \{1,2,...,s\}$. Let $\tau^{(e)}(n)$ denote the number of exponential divisors of $n$, where $\tau^{(e)}(1)=1$ by convention. The aim of the present paper is to establish an asymptotic formula with remainder term for the $r$-th power of the function $\tau^{(e)}$, where $r\ge 1$ is an integer. This improves an earlier result of {\sc M. V. Subbarao} [5].

Integers with a large smooth divisor

Abstract: We study the function ! (x,y,z) that counts the number of positive integers n ! x which have a divisor d > z with the property that p ! y for every prime p dividing d. We also indicate some cryptographic applications of our results.

Digital filters for semiconductor devices

Abstract: A memory device that, in certain embodiments, includes a memory element and a digital filter. The digital filter may include a counter and a divider, where the divider is configured to divide a count from the counter by a divisor.

The number of representations by sums of squares and triangular numbers

Abstract: In this paper, we present eighteen interesting infinite products and their Lambert series expansions. From these, we deduce formulae for the number of representations of an integer n by eighteen quadratic forms in terms of divisor sums.

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.

A criterion for the logarithmic differential operators to be generated by vector fields

Abstract: We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is gen- erated by logarithmic vector fields. We give • a sufficient criterion for the property, • a simple proof of F.J. Calderon-Moreno's theorem that free divisors have the property, • a proof that divisors in dimension 3 with only isolated quasi-homogeneous singularities have the property, • an example of a non-free divisor with non-isolated singularity having the property, • an example of a divisor not having the property, and • an algorithm to compute the V-filtration along a divisor up to a given order.

The distribution of integers with at least two divisors in a short interval

Abstract: We estimate the density of integers which have more than one divisor in an interval (y, z] with z y + y/(log y)log 4 ? 1. As a consequence, we determine the precise range of z such that most integers which have at least one divisor in (y, z] have exactly one such divisor.

Affine Algebraic Varieties

Abstract: In this paper, we give new criteria for affineness of a variety defined over $\Bbb{C}$. Our main result is that an irreducible algebraic variety $Y$ (may be singular) of dimension $d$ ($d\geq 1$) defined over $\Bbb{C}$ is an affine variety if and only if $Y$ contains no complete curves, $H^i(Y, {\mathcal{O}}_Y)=0$ for all $i>0$ and the boundary $X-Y$ is support of a big divisor, where $X$ is a projective variety containing $Y$. We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor $D$ and the affineness of $Y$. If $Y$ is an affine variety, then the ring $\Gamma (Y, {\mathcal{O}}_Y)$ is noetherian. However, to prove that $Y$ is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring $\Gamma (Y, {\mathcal{O}}_Y)$ directly but use the techniques of sheaf and cohomology.

Stability of the divisor class group upon completion

Abstract: If $A$ is a local, normal approximation domain with finite divisor class group, then $Cl(A) \cong Cl(\widehat{A})$.