Results 1 to 10 of about 41,624 (236)
On a Sum Involving the Sum-of-Divisors Function [PDF]
Journal of Mathematics, 2021 Let σn be the sum of all divisors of n and let t be the integral part of t. In this paper, we shall prove that ∑n≤xσx/n=π2/6x log x+Oxlog x2/3log2 x4/3 for x⟶∞, and that the error term of this asymptotic formula is Ωx.
Feng Zhao, Jie Wu
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The sum of divisors of a quadratic form [PDF]
Acta Arithmetica, 2014 We study the sum of divisors of the quadratic form $m_1^2+m_2^2+m_3^2$. Let $$S_3(X)=\sum_{1\le m_1,m_2,m_3\le X}\tau(m_1^2+m_2^2+m_3^2).$$ We obtain the asymptotic formula $$S_3(X)=C_1X^3\log X+ C_2X^3+O(X^2\log^7 X),$$ where $C_1,C_2$ are two constants.
Zhao, Lilu
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Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
Open Mathematics, 2017 The convolution sum, ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms
Ntienjem Ebénézer
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INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH, 2023 We show that the recurrence relation deduced by Robbins and Osler et al for the sum of divisors function can be solved in terms of the complete Bell polynomials.
R. Sivaraman, J. Bulnes, J. L. Bonilla
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Quasi-modularity of generalized sum-of-divisors functions [PDF]
Research in Number Theory, 2015 In 1919, P. A. MacMahon studied generating functions for generalized divisor sums. In this paper, we provide a framework in which to view these generating functions in terms of Jacobi forms, and prove that they are quasi-modular forms.Comment: 11 ...
Rose, Simon
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Electronic Proceedings in Theoretical Computer Science, 2015 A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum.
John Cowles, Ruben Gamboa
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MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms [PDF]
, 2010 We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their ...
Andrews, George E., Rose, Simon CF
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Corrigendum to: "Some modular considerations regarding odd perfect numbers – Part II" [Notes on Number Theory and Discrete Mathematics, 2020, Vol. 26, No. 3, 8–24] [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 In [2], the authors proposed a theorem which they recently found out to contradict Chen and Luo's results [1]. In the present paper, we provide the correct form of this theorem.
Jose Arnaldo Bebita Dris +1 more
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ON A SUM INVOLVING CERTAIN ARITHMETIC FUNCTIONS ON PIATETSKI–SHAPIRO AND BEATTY SEQUENCES
Проблемы анализа, 2022 Let 𝑐, 𝛼, 𝛽 ∈ R be such that ...
T. Srichan
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On a modification of Set(n) [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 A modification of the set Set(n) for a fixed natural number n is introduced in the form: Set(n, f), where f is an arithmetic function. The sets Set(n,φ), Set(n,ψ), Set(n,σ) are discussed, where φ, ψ and σ are Euler's function, Dedekind's function and the
Krassimir T. Atanassov, József Sándor
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