Divisor Problem Research Articles

Approximate Integer Common Divisor Problem Relates to Implicit Factorization

In this paper, we analyze how to calculate the GCD of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ( ≥ 2) many large integers, given their approximations. This problem is known as the approximate integer common divisor problem in literature. Two versions of the problem, presented by Howgrave-Graham in CaLC 2001, turn out to be special cases of our analysis when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> = 2. We relate the approximate common divisor problem to the implicit factorization problem as well. The later was introduced by May and Ritzenhofen in PKC 2009 and studied under the assumption that some of Least Significant Bits (LSBs) of certain primes are the same. Our strategy can be applied to the implicit factorization problem in a general framework considering the equality of (i) most significant bits (MSBs), (ii) least significant bits (LSBs), and (iii) MSBs and LSBs together. We present new and improved theoretical as well as experimental results in comparison with the state of the art work in this area.

On a divisor problem related to the Epstein zeta-function, II

Recently by using the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and of Lü, and to establish corresponding Ω-estimates.

A new proof of the Voronoï summation formula

We present a short alternative proof of the Voronoï summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula for a Schrödinger operator on a noncompact quantum graph.

An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of δ-like potentials with strength κ > 0 on the half line and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval In = [0, ln], , to each edge with length . We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each κ > 0 and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by λn = n2, , with multiplicities d(n), where d(n) denotes the divisor function. We can thus relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics for the eigenvalue counting function. Based on an exact trace formula, the Voronoï summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.

New estimates of the remainder in an asymptotic formula in the multidimensional Dirichlet divisor problem

We obtain a new value of the Karatsuba constant in the multidimensional Dirichlet divisor problem. We also find a new value of the exponent of the main parameter in the estimate of the mean value of the remainder in a given asymptotics. The proof of the main statements is based on the derivation of a new estimate of the Carleson abscissa in the theory of the Riemann zeta function.

A quasianalyticity property for monogenic solutions of small divisor problems

We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K_j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K_j's.

ON THE FIFTH-POWER MOMENT OF Δ(x)

Let Δ(x) denote the error term in the Dirichlet divisor problem. In this paper, we study the fifth-power moment of Δ(x) and prove that [Formula: see text] with δ5 = 3/80, which improves the previous results.

Time-Periodic Linearized Solutions of the Compressible Euler Equations and a Problem of Small Divisors

It has been unknown since the time of Euler whether or not time-periodic sound wave propagation is physically possible in the compressible Euler equations, due mainly to the ubiquitous formation of shock waves. The existence of such waves would confirm the possibility of dissipation free long distance signaling. Following our work in [B. Temple and R. Young, A paradigm for time-periodic sound wave propagation in the compressible Euler equations, Methods Appl. Anal., 16 (2009), pp. 341–363], we derive exact linearized solutions that exhibit the simplest possible periodic wave structure that can balance compression and rarefaction along characteristics in the nonlinear Euler problem. These linearized waves exhibit interesting phase and group velocities analogous to linear dispersive waves. Moreover, when the spatial period is incommensurate with the time period, the sound speed is incommensurate with the period, and a new periodic wave pattern is observed in which the sound waves move in a quasiperiodic trajectory though a periodic configuration of states. This establishes a new way in which nonlinear solutions that exist arbitrarily close to these linearized solutions can balance compression and rarefaction along characteristics in a quasiperiodic sense. We then rigorously establish the spectral properties of the linearized operators associated with these linearized solutions. In particular we show that the linearized operators are invertible on the complement of a one-dimensional kernel containing the periodic solutions only in the case when the wave speeds are incommensurate with the periods, but these invertible operators have small divisors, analogous to KAM theory. Almost everywhere algebraic decay rates for the small divisors are proven. In particular this provides a solid framework for the problem of perturbing these linearized solutions to exact nonlinear periodic solutions of the fully compressible Euler equations.

The divisor problem for binary cubic forms

We investigate the average order of the divisor function at values of binary cubic forms that are reducible over ℚ and discuss some applications.

Inverse elementary divisor problems for nonnegative matrices

Inverse elementary divisor problems for nonnegative matrices

Higher moments of the error term in the divisor problem

It is proved that, if k ≥ 2 is a fixed integer and 1 ≪ H ≤ (1/2)X, then $$ \int_{X - H}^{X + H} {\Delta _k^4 \left( x \right) } dx \ll _\varepsilon X^\varepsilon \left( {HX^{{{\left( {2k - 2} \right)} \mathord{\left/ {\vphantom {{\left( {2k - 2} \right)} k}} \right. \kern-\nulldelimiterspace} k}} + H^{{{\left( {2k - 3} \right)} \mathord{\left/ {\vphantom {{\left( {2k - 3} \right)} {\left( {2k + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2k + 1} \right)}}} X^{{{\left( {8k - 8} \right)} \mathord{\left/ {\vphantom {{\left( {8k - 8} \right)} {\left( {2k + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2k + 1} \right)}}} } \right), $$ where Δk(x) is the error term in the general Dirichlet divisor problem. The proof uses a Voronoi-type formula for Δk(x), and the bound of Robert-Sargos for the number of integers when the difference of four kth roots is small. The size of the error term in the asymptotic formula for the mth moment of Δ2(x) is also investigated.

Small Divisor Problem in Dynamical Systems and Analytic Solutions of a q-Difference Equation with a Singularity at the Origin

In this paper, we consider a q-difference equation $$\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(y(q^jz))^{t}=G(z)$$ in the complex field \({\mathbb C,}\) where Ct,j(z) and G(z) have a h1 order pole and a h2 order pole at z = 0, respectively. Under the case 0 < |q| < 1 or |q| = 1, we give the existence of local analytic solutions for the above equation by using small divisor theory in dynamical systems.

Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line. This article is a survey of recent developments on the research of these famous error terms in number theory. These include upper bounds, Ω-results, sign changes, moments and distribution, etc. A few open problems are also discussed.

A note on the mean square of |ζ(½ + it )|

Using the method of Bombieri and Iwaniec, new upper bounds are obtained for the absolute value of exponential sums S =  H/2<hH  M/2<mM exp  2�IiTF  m + h M  . 2�IiTF  m . h M  with parameters T and M large, T1/3 <M < T2/3 and 1  H <MT .17/57, and with certain non-vanishing conditions imposed on the derivatives of the function F : [1/3, 3] �¨ R. For E(T), the error term in the asymptotic formula for the mean square of |�A(1/2 + it)|, one consequently obtains a new bound E(T)  T�AE log�O(T + 2), with a constant �O < 4, and with the same �emain exponent�f, �AE = 131/416, as occurs in the sharpest upper bounds yet found for the error terms of the circle and divisor problems. The proofs utilize recent advances by Huxley, applying to the Bombieri.Iwaniec method in general, and independent progress of the author on a question specific to the sum S. Conditional improvements are given, subject to Huxley�fs �eHypothesis H(�E, �E)�f concerning the number of integer points near a given bounded curve.

On higher-power moments of the error term for the divisor problem with congruence conditions

For a positive integer n, the divisor function with congruence conditions d(n; l 1, M 1, l 2, M 2) denotes the number of factorizations n = n 1 n 2, where each of the factors \({n_i\in\mathbb{N}}\) belongs to a prescribed congruence class l i modulo M i (i = 1, 2). In this paper we study the higher power moments of the error term in the asymptotic formula of \({\sum\nolimits_{n\leq M_1M_2x}d(n;l_1,M_1,l_2,M_2)}\) .

Analytic properties of Dirichlet series obtained from the error term in the Dirichlet divisor problem

Analytic properties of Dirichlet series obtained from the error term in the Dirichlet divisor problem

On a divisor problem related to the Epstein zeta-function

In 1995, Sankaranarayanan studied a divisor problem related to the Epstein zeta-function. By the theory of modular forms and the Riemann zeta-function, we are able to improve his result for a number of cases.

On the Existence of Quasipattern Solutions of the Swift–Hohenberg Equation

Quasipatterns (two-dimensional patterns that are quasiperiodic in any spatial direction) remain one of the outstanding problems of pattern formation. As with problems involving quasiperiodicity, there is a small divisor problem. In this paper, we consider 8-fold, 10-fold, 12-fold, and higher order quasipattern solutions of the Swift–Hohenberg equation. We prove that a formal solution, given by a divergent series, may be used to build a smooth quasiperiodic function which is an approximate solution of the pattern-forming partial differential equation (PDE) up to an exponentially small error.

Explicit representations of the integral containing the error term in the divisor problem

We shall investigate several properties of the integral $$ \int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt} $$ with a natural number k, a non-negative integer j and a complex variable θ, where Δ k (x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.

Dirichlet series obtained from the error term in the Dirichlet divisor problem

In this paper we consider the Dirichlet series obtained from the error term in the Dirichlet divisor problem. We shall prove their analytic continuations and determine the locations of poles. We shall also discuss the relation between our results and the conjecture of Lau and Tsang on the mean square estimate of the error term.