Number Of Divisors Research Articles

Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums [Formula: see text]where [Formula: see text] is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to [Formula: see text], do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.

On certain restricted partition functions

The objective of this paper is to establish explicit formulas for certain restricted partition function in terms of divisor function

Convolution sums of restricted divisor functions derived from Dirichlet convolution

Convolution sums of restricted divisor functions derived from Dirichlet convolution

Study of Dynamic Modulus of Asphalt Mix after Reinforcement of Sandstone

Sandstone has poor mechanical properties. To facilitate the application of sandstone into asphalt mixtures, sandstone was treated by immersion in sodium silicate solution, and the dynamic modulus after reinforcement was used as a criterion. The results showed that the mechanical properties of the sandstone aggregate treated with sodium silicate were improved, and the dynamic modulus was increased by 18.2%, which will help to reduce rutting. The dynamic modulus and phase angle can be effectively predicted over a wide frequency range using the sigma function and the Kramers–Kronig relationship. Sandstone asphalt mixtures basically conform to linear viscoelasticity, but the phase angle changes are more complicated at high temperatures and do not vary monotonically with frequency. By calculating the rutting coefficient, fatigue coefficient, and DSRFn parameters for performance prediction, it was found that an increase in dynamic modulus resulted in a significant increase in the rutting coefficient but a decrease in the cracking resistance.

On some geometric properties of sequence spaces of generalized arithmetic divisor sum function

Recently, some new sequence spaces ℓp(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}(\\mathfrak{A}^{\\alpha })$\\end{document}(0<p<∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(0< p<\\infty )$\\end{document}, c0(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{0}(\\mathfrak{A}^{\\alpha })$\\end{document}, c(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c(\\mathfrak{A}^{\\alpha })$\\end{document}, and ℓ∞(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{\\infty }(\\mathfrak{A}^{\\alpha })$\\end{document} have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of Aα=(an,vα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{A}^{\\alpha }=(a_{n,v}^{\\alpha })$\\end{document}, where am,vα={vαρ(α)(m),v∣m,0,v∤m,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ a_{\\mathfrak{m},v}^{\\alpha }=\\left \\{ \ extstyle\\begin{array}{c@{\\quad}c@{\\quad}c} \\dfrac{v^{\\alpha }}{\\rho ^{(\\alpha )}(\\mathfrak{m})} & , & v\\mid \\mathfrak{m}, \\\\ 0 & , & v\ mid \\mathfrak{m},\\end{array}\\displaystyle \\right . $$\\end{document}and ρ(α)(m):=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho ^{(\\alpha )}(\\mathfrak{m}):=$\\end{document} sum of the αth\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha ^{\ ext{th}}$\\end{document} power of the positive divisors of m∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{m}\\in \\mathbb{N}$\\end{document}. They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes.This article deals with some geometric properties of these sequence spaces.

Nonlinear Schrödinger equation in terms of elliptic and hyperelliptic σ functions

Abstract It is known that the elliptic function solutions of the nonlinear Schrödinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, [ − i ∂ ∂ t + α ∂ ∂ u ] Ψ − 1 2 ∂ 2 ∂ u 2 Ψ + ( Ψ ∗ Ψ ) Ψ = 1 2 ( 2 β + α 2 − 3 ℘ ( v ) ) Ψ , where Ψ ( u ; v , t ) := e α u + i β t + c e − ζ ( v ) u σ ( u + v ) σ ( u ) σ ( v ) , its dual Ψ ∗ ( u ; v , t ) , and certain complex numbers α , β and c. In this paper, we generalize the algebraic differential relation to those of genera two and three in terms of the hyperelliptic sigma functions.

Sign Changes of the Error Term in the Piltz Divisor Problem

Abstract We study the function $\Delta _{k}(x):=\sum _{n\leq x} d_{k}(n) - \mbox{Res}_{s=1} ( \zeta ^{k}(s) x^{s}/s )$, where $k\geq 3$ is an integer, $d_{k}(n)$ is the $k$-fold divisor function, and $\zeta (s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindelöf hypothesis (LH) is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon }$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon }$, such that $|\Delta _{k}(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. In particular, $\Delta _{k}(x)$ does not change sign in any of these subintervals. If the Riemann hypothesis (RH) is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^{2}-2}$. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of $\Delta _{k}(x+h)-\Delta _{k}(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $\Delta _{k}(x)$, which we obtain by combining a method of Tsang with a technique of Lester.

The Riemann zeta function and exact exponential sum identities of divisor functions

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities involving the divisor functions, and we provide examples of these identities. We conjecturally propose a method to compute divisor functions by matrix inversion, without employing arithmetic techniques.

Euler’s partition function in terms of 2-adic valuation

The 2-adic valuation of an integer n is the exponent of the highest power of 2 that divides n and is denoted by ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. In this paper, we prove that Euler’s partition function p(n) can be expressed in terms of ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. Our approach allows us to express the sum of positive divisors of n in terms of ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. We introduce the notion of 2-adic color partition and provide a new combinatorial interpretation for Euler partition function p(n). Connections between partitions and the game of m-Modular Nim with two heaps are presented in this context.

A segment of Euler product associated to a certain Dirichlet series

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function σα(n):=∑d|ndα. We obtain an exact identity relating the Dirichlet series ζ(s)ζ(s−α) and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip.

Sums of Higher Divisor Functions with Almost Equal Variables

Sums of Higher Divisor Functions with Almost Equal Variables

The Weil bound for generalized Kloosterman sums of half-integral weight

Abstract Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

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 certain bounds for the divisor function

We offer various bounds for the divisor function d(n), in terms of n, or other arithmetical functions.

Monotone Nondecreasing Sequences of the Euler Totient Function

Let M(x) denote the largest cardinality of a subset of {n∈N:n≤x}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{n \\in \\mathbb {N}: n \\le x\\}$$\\end{document} on which the Euler totient functionφ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (n)$$\\end{document} is nondecreasing. We show that M(x)=1+O(loglogx)5logxπ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(x) = \\left( 1+O\\left( \\frac{(\\log \\log x)^5}{\\log x}\\right) \\right) \\pi (x)$$\\end{document} for all x≥10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\ge 10$$\\end{document}, answering questions of Erdős and Pollack–Pomerance–Treviño. A similar result is also obtained for the sum of divisors function σ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma (n)$$\\end{document}.

From sums of divisors to partition congruences

Let z be a complex number. For any positive integer n it is well known that the sum of the zth powers of the positive divisors of n can be computed without knowing all the divisors of n, if we take into account the factorization of n. In this paper, we rely on the integer partitions of n in order to investigate computational methods for ∑d|n(±1)d+1dz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (\\pm 1)^{d+1}\\,d^z$$\\end{document}, ∑d|n(-1)n/d+1dz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (-1)^{n/d+1}\\,d^z$$\\end{document} and ∑d|n(-1)n/d+ddz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum _{d|n} (-1)^{n/d+d}\\,d^z$$\\end{document}. To compute these sums of divisors of n, it is sufficient to know the multiplicity of 1 in each partition involved in the computational process. Our methods do not require knowing the divisors of n or the factorization of n. New congruences involving Euler’s partition function p(n) are experimentally discovered in this context.

On mean values for the exponential sum of divisor functions

In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris 360 (2022) 419–424].

Robin inequality for n/phi(n)

Let $\varphi(n)$ be the Euler function, $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function and $\gamma=0.577\ldots$ the Euler constant. In 1982, Robin proved that, under the Riemann hypothesis, $\sigma(n)/n &lt; e^\gamma \log\log n$ holds for $n &gt; 5040$ and that this inequality is equivalent to the Riemann hypothesis. The aim of this paper is to give a similar equivalence for $n/\varphi(n)$.

The circle method and shifted convolution sums involving the divisor function

Let Q(x) be a positive definite integral quadratic form with the determinant D being squarefree, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d(n) and Fourier coefficients r(n,Q). With more efforts, our method should have a number of applications for other multiplicative functions.

Trigonometric analogues of the identities associated with twisted sums of divisor functions

Trigonometric analogues of the identities associated with twisted sums of divisor functions