Number Of Divisors Research Articles

On Evaluation of Convolution Sums Involving Divisor Functions and Partition Functions

On Evaluation of Convolution Sums Involving Divisor Functions and Partition Functions

Bounds on bilinear forms with Kloosterman sums

AbstractWe prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet ‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.

Correction: Komeda et al. Algebraic Construction of the Sigma Function for General Weierstrass Curves. Mathematics 2022, 10, 3010

The authors wish to make the following corrections to this paper [...]

Bressoud–Subbarao Type Weighted Partition Identities for a Generalized Divisor Function

Bressoud–Subbarao Type Weighted Partition Identities for a Generalized Divisor Function

Generalized Euler's \( \Phi_w \)-function and the divisor sum \(T_{k_w} \)-function of edge weighted graphs

In this work, generalized Euler's \(\Phi_w \)-function of edge weighted graphs is defined which consists of the sum of the Euler's \(\varphi\)-function of the weight of edges of a graph and we denote it by \(\Phi_w(G)\) and the general form of Euler's \(\Phi_w\)-function of some standard edge weighted graphs is determined. Also, we define the divisor sum \(T_{k_w}\)-function \(T_{k_w}(G)\) of the graph \(G\), which is counting the sum of the sum of the positive divisor \(\sigma_k \)-function for the weighted of edges of a graph \(G\). It is determined a relation between generalized Euler's \(\Phi_w\)-function and generalized divisor sum \(T_{k_w}\)-function of edge weighted graphs.

Identities associated to a generalized divisor function and modified Bessel function

Identities associated to a generalized divisor function and modified Bessel function

On the universal p-adic sigma and Weierstrass zeta functions

For primes p>3 we produce a new derivation of the universal p-adic sigma function and p-adic Weierstrass zeta functions of Mazur and Tate for elliptic curves with good ordinary or multiplicative reduction by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.

Triple correlations of ternary divisor functions. II

Let [Formula: see text] denote the ternary divisor function. By introducing a simple argument, we prove an asymptotic formula for the triple correlation [Formula: see text] averaging over a short variable [Formula: see text], with an error term saving an arbitrary power of the logarithm over the trivial bound. We also prove an asymptotic formula by taking one of the ternary divisor functions to be the divisor function [Formula: see text], which substantially improves our previous results.

Runs of Consecutive Integers Having the Same Number of Divisors

Our objective is to provide an upper bound for the length ℓN of the longest run of consecutive integers smaller than N which have the same number of divisors. We prove in an elementary way that log ℓN << (log N log log N)λ, where λ=1/2. Using estimates for the Jacobsthal function, we then improve the result to λ=1/3.

On fractional sums of the divisor functions

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by [O. Bordellés, On certain sums of number theory, Int. J. Number Theory 18(9) (2022) 2053–2074; K. Liu, J. Wu and Z. S. Yang, On some sums involving the integral part function, preprint (2021); arXiv:2109.01382v1 [math.NT]]. Precisely, we can show that [Formula: see text] where [Formula: see text] is an arbitrary small positive constant and [Formula: see text] is the number of representations of [Formula: see text] as product of [Formula: see text] natural numbers.

SUMMATION OF THE FOURIER COEFFICIENTS OF THE NON-HOLOMORPHIC EISENSTEIN CUSP SERIES OF THE PSL(2,Z) GROUP

The Fourier coefficients of the non-holomorphic Eisenstein cusp forms of the Laplace-Beltrami operator are here summed. The summation is performed after the Dirichlet series related to the divisor function, after the properties of the meromorphic continuability; a new dependence on the Euler’s γ constant is also found.

Değiştirilmiş tek bölen fonksiyonları ve bölen yaprak modeli

In this research, modelling of the leaves with the help of divisor functions are worked on. Given a positive integer k (1 ≤ k ≤ 100), we investigate solutions of the equation σ(n) = σ(n + 2k) with odd square-free integer n. Further, for a positive integer l and odd prime q, there are no results of the equation σ2i(n) = σ2i(q). As an application, we pose the basic structure of the leaves model for real-time virtual ecosystem construction derived from the equation of shifted odd divisor functions. Also, the elliptic, flabellate and five-lobes leaves’s area and the growth process of the leaves were made modelling with the help of divisor functions.

Mathcal{N} = 2* Schur indices

We find closed-form expressions for the Schur indices of 4d mathcal{N} = 2* super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.

On power values of sum of divisors function in arithmetic progressions

Let $$a\ge 1, b\ge 0$$ and $$k\ge 2$$ be any given integers. It has been proven that there exist infinitely many natural numbers m such that sum of divisors of m is a perfect kth power. We try to generalize this result when the values of m belong to any given infinite arithmetic progression $$an+b$$ . We prove if a is relatively prime to b and order of b modulo a is relatively prime to k then there exist infinitely many natural numbers n such that sum of divisors of $$an+b$$ is a perfect kth power. We also prove that, in general, either sum of divisors of $$an+b$$ is not a perfect kth power for any natural number n or sum of divisors of $$an+b$$ is a perfect kth power for infinitely many natural numbers n.

Теоретико-числовi функцiї для цiлих ґаусових чисел

The classical number-theoretic functions – a number of divisors τ(n), sum of the divisors σ(n) and product of the divisors π(n) of a positive integer n – were generalized to the ring Z[i] of Gaussian integers. For the evaluation of the corresponding functions τ*(α), σ*m(α) and π*(α), obtained were the explicit formulae that use the canonical representation of α. A number of properties of these functions were studied, in particular, estimates from above for the functions τ*(α) and σ*m(α) and the properties connected with divisibility of their values by certain numbers. Researched are also sums of products of powers of the divisors for α∈Z[i].

On the divisors of natural and happy numbers: a study based on entropy and graphs

<abstract><p>The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences.</p></abstract>

On the divisors of natural and happy numbers: a study based on entropy and graphs

<abstract><p>The features of numerical sequences and time series have been studied by using entropies and graphs. In this article, two sequences derived from the divisors of natural numbers are investigated. These sequences are obtained either directly from the divisor function or by recursively applying the divisor function. For comparison purposes, analogous sequences formed from the divisors of happy numbers are also examined. Firstly, the informational entropy of these four sequences is numerically determined. Then, each sequence is mapped into graphs by employing two visibility algorithms. For each graph, the average degree, the average shortest-path length, the average clustering coefficient, and the degree distribution are calculated. Also, the links in these graphs are quantified in terms of the parity of the numbers that these links connect. These computer experiments suggest that the four analyzed sequences exhibit characteristics of quasi-random sequences.</p></abstract>

Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,\theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

On the values of Weierstrass zeta and sigma functions (with an appendix by David Masser)

On the values of Weierstrass zeta and sigma functions (with an appendix by David Masser)

A NOTE ON A CLASSICAL CONNECTION BETWEEN PARTITIONS AND DIVISORS

In this note, we consider the number of k’s in all the partitions of n in order to provide a new proof of a classical identity involving Euler’s partition function p(n) and the sum of the positive divisors function a(n). New relations connecting classical functions of multiplicative number theory with the partition function p(n) from additive number theory are introduced in this context. The fascinating feature of these relations is their common nature. A new identity for the number of 1’s in all the partitions of n is derived in this context.