Divisor Problem Research Articles

Convolution of periodic multiplicative functions and the divisor problem

Convolution of periodic multiplicative functions and the divisor problem

The general divisor problem of higher moments of coefficients attached to the Dedekind zeta function

The general divisor problem of higher moments of coefficients attached to the Dedekind zeta function

Toward optimal exponent pairs

We quantify the set of known exponent pairs ( k , ℓ ) (k, \ell ) and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds of the Riemann zeta-function ζ ( s ) \zeta (s) in the critical strip, estimates of the moments of ζ ( 1 / 2 + i t ) \zeta (1/2 + it) and the generalised Dirichlet divisor problem.

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

Prime and Möbius correlations for very short intervals in $\fq[x

abstract: We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in ``very short intervals'' of the form $I(f):=\{f(x) + a : a \in\fp\}$ for $f(x)\in\fp[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\"{o}bius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for ``generic'' set of shifts. We can also exhibit examples for which there is no cancellation at all in M\"{o}bius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\"{o}bius sums is {\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic ``primes are independent'' fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\"{o}bius $\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated). We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x)\in\fq[x]$ and intervals of the form $f(x)+a$ for $a\in\fq$, where $p$ is fixed and $q=p^{l}$ grows.

On the generalised Dirichlet divisor problem

AbstractWe improve unconditional estimates on , the remainder term of the generalised divisor function, for large . In particular, we show that for all sufficiently large fixed .

On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function $\zeta_{\mathbb{K}_3}(s)$ of a non-normal cubic extension $\mathbb{K}_3$ of $\mathbb{Q}$ over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\[ \sum_{{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}\leq {x}}\atop{(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\in\mathbb{Z}^{6}}}a_{k,\mathbb{K}_3} (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}+a_{6}^{2}),\]where $(\zeta_{\mathbb{K}_3}(s))^{k}:=\sum_{n=1}^{\infty}\frac{a_{k,\mathbb{K}_3}(n)}{n^{s}}$.

AIOL: An Improved Orthogonal Lattice Algorithm for the General Approximate Common Divisor Problem

The security of several fully homomorphic encryption (FHE) schemes depends on the intractability assumption of the approximate common divisor (ACD) problem over integers. Subsequent efforts to solve the ACD problem as well as its variants were also developed during the past decade. In this paper, an improved orthogonal lattice (OL)-based algorithm, AIOL, is proposed to solve the general approximate common divisor (GACD) problem. The conditions for ensuring the feasibility of AIOL are also presented. Compared to the Ding–Tao OL algorithm, the well-known LLL reduction method is used only once in AIOL, and when the error vector r is recovered in AIOL, the possible difference between the restored and the true value of p is given. Experimental comparisons between the Ding-Tao algorithm and ours are also provided to validate our improvements.

Divisor Problem for the Greatest Common Divisor of Integers in Piatetski-Shapiro and Beatty Sequences

In this paper, we derive several asymptotic formulas for the sum of d(gcd(m,n)), where d(n) is the divisor function and m,n are in Piatetski-Shapiro and Beatty sequences.

Fully Homomorphic Encryption Scheme Over Integers Based on DGHV Scheme

Homomorphic encryption was introduced to enable untrusted parties to apply computation over encrypted data, without decrypting the message. It makes a promising future to solve security challenges with Cloud Computing. Still, because of complex problems, HE is not yet usable in real-world applications. However homomorphic encryption suffers from many complex issues like large-time computation, which is difficult to implement, and huge ciphertext size. In this paper, we take into account many vital homomorphic encryption schemes, all of them is depend on computing over integers, by converting the message to a binary format, then the ciphertext is calculated to every single bit of plaintext. So, we suggested another symmetric Fully Homomorphic encryption that depends on an integer number without converting it to binary format, it encrypts the message using a prime secret key. The proposed system is considered a great improvement to the DGHV system. Where it reduced the ciphertext size and the execution time significantly. The present proposed system shows an improvement of DGHV where it is faster about 23.8 times from DGHV, and we gain one ciphertext for the whole message instead of ciphertext for every bit of the message. The scheme's hardness depends on the greatest common divisor (GCD) problem. We present a detailed explanation of these schemes showing their efficient techniques and compare it with the proposed algorithm in terms of ciphertext size of algorithm, and execution time and analyze their security them finally draw several conclusions.

Construction of similarity measure for intuitionistic fuzzy sets and its application in face recognition and software quality evaluation

As a generalization of fuzzy sets, intuitionistic fuzzy sets (IFSs) are more capable of representing and addressing uncertainty in real-world problems. As a result, IFSs have been utilized in various areas of application. However, the distance and similarity measures between the two IFSs are still an open issue that has drawn much attention over the past few decades. Even though several intuitionistic fuzzy similarity measures (IFSMs) have been developed, a number of issues still exist, including counter-intuitive results, ‘the zero divisor problem,’ violation of similarity measure axioms, and being incompetent to detect minor changes in membership or non-membership. To overcome these shortcomings, a novel intuitionistic fuzzy similarity measure (IFSM) has been introduced in this study. Unlike existing measures, this method considered the global maximum and minimum of differences in memberships and differences in non-memberships, along with their individual differences, to construct an IFSM. Moreover, it has been shown that a convex combination of two similarity measures is also a similarity measure. Some numerical examples are employed to emphasize the advantages and strengths of the proposed method over existing ones. The suggested IFSM has been implemented on a few pattern classification issues to demonstrate its efficacy and a new face recognition method is also presented. Moreover, an entropy measure is introduced using the proposed IFSM, which is further used to construct the weights of attributes in the MADM method. Furthermore, the MADM technique, IF-E-TOPSIS, is constructed using the suggested IFSM and entropy measure. The effectiveness of this method is shown by utilizing it for software quality assessment, and the results are compared with existing ones to highlight the superiority of the proposed method.

The Dirichlet divisor problem over square-free integers and unitary convolutions

We obtain an asymptotic formula for the sum $\tilde{D}_2$ of the divisors of all square-free integers less than or equal to $x$, with error term $O(x^{1/2 + \epsilon})$. This improves the error term $O(x^{3/4 + \epsilon})$ presented in [7] obtained via analytical methods. Our approach is elementary and it is based on the connections between the function $\tilde{D}_2$ and unitary convolutions.

Hopf-like bifurcation in a wave equation at a removable singularity

It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and accumulates to zero. The upshot is that the singularity of the linearized operator at criticality which stems from the well known small divisor problem for the wave operator, is entirely removed without the need to exclude parameters via Diophantine conditions, nor the use of accelerated convergence schemes. Only the contraction mapping principle is used.

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

Karatsuba's divisor problem and related questions

Доказано, что $$ \sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}}, \qquad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ где $\tau(n)=\sum_{d\mid n}1$ - количество делителей числа $n$, а суммирование в первой сумме ведется по простым числам. Библиография: 14 названий.

Karatsuba's divisor problem and related questions

We prove that $$ \sum_{p \leq x} \frac{1}{\tau(p-1)} \asymp \frac{x}{(\log x)^{3/2}} \quadand\quad \sum_{n \leq x} \frac{1}{\tau(n^2+1)} \asymp \frac{x}{(\log x)^{1/2}}, $$ where $\tau(n)=\sum_{d\mid n}1$ is the number of divisors of $n$, and the first sum is taken over prime numbers. Bibliography: 14 titles.

An induction principle for the Bombieri-Vinogradov theorem over [formula omitted] and a variant of the Titchmarsh divisor problem

Let Fq[t] be the polynomial ring over the finite field Fq. For arithmetic functions ψ1,ψ2:Fq[t]→C, we establish that if a Bombieri-Vinogradov type equidistribution result holds for ψ1 and ψ2, then it also holds for their Dirichlet convolution ψ1⁎ψ2. As an application of this, we resolve a version of the Titchmarsh divisor problem in Fq[t]. More precisely, we obtain an asymptotic for the average behaviour of the divisor function over shifted products of two primes in Fq[t].

A Symmetric Form of the Mean Value Involving Non-Isomorphic Abelian Groups

Let a(n) be the number of non-isomorphic abelian groups of order n. In this paper, we study a symmetric form of the average value with respect to a(n) and prove an asymptotic formula. Furthermore, we study an analogue of the well-known Titchmarsh divisor problem involving a(n).

On Wigert's type divisor problem

In this paper we study a special kind of divisor function which was first introduced by Wigert while investigating a series involving k-th power divisor function. We will give a sharper estimate of the error term appeared in the asymptotic formula for the above mentioned divisor function. Furthermore, we will give an account of the mean value of the error term as well as mean square of the error term.

Notes on general divisor problems related to Maass cusp forms

Notes on general divisor problems related to Maass cusp forms