Results 31 to 40 of about 31,249 (173)
The Nicolas and Robin inequalities with sums of two squares [PDF]
, 2007 In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\sigma(n)5040$, where $\sigma(n)$ is the sum of divisors function, and $\gamma$ is the Euler-Mascheroni constant.
Banks, William D. +3 more
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Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica, 2014 We obtain an arithmetic proof and a refinement of the inequality ϕ (nk) + σk(n) < 2nk, where n ≧ 2 and k ≧ 2. An application to another inequality is also provided.
Sándor József
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DISTRIBUTION OF VALUES OF THE SUM OF UNITARY DIVISORS IN RESIDUE CLASSES
Проблемы анализа, 2016 In this paper we prove the tauberian type theorem containing the asymptotic series for the Dirichlet series. We use this result to study distribution of sum of unitary divisors in residue classes coprime with a module.
B. M. Shirokov, L. A. Gromakovskaya
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Congruences and Exponential Sums with the Euler Function [PDF]
, 2004 This is a preprint of a book chapter published in High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications, AMS (2004).
Banks, William David, 1964- +1 more
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An analogue in certain unique factorization domains of the Euclid-Euler theorem on perfect numbers
International Journal of Mathematics and Mathematical Sciences, 1990 We show that there exists a natural extention of the sum of divisors function to all unique factorization domains F having a finite number of units such that if a perfect number in F is defined to be an integer η whose proper divisors sum to η, then the ...
Wayne L. McDaniel
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Quasi-modularity of generalized sum-of-divisors functions [PDF]
Research in Number Theory, 2015 11 ...
openaire +4 more sources
On a function related to Ramanujan's Tau function
International Journal of Mathematics and Mathematical Sciences, 1985 For the function ψ=ψ12, defined by ∑1∞ψ(n)xn=x∏1∞(1−x2n)12 (|x|
John A. Ewell
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Divisibility and sequence properties of σ⁺ and φ⁺ [PDF]
Notes on Number Theory and Discrete MathematicsInspired by Lehmer’s and Deaconescu’s conjectures, as well as various analogue problems concerning Euler’s totient function φ(n), Schemmel’s totient function S₂(n), Jordan totient function Jₖ, and the unitary totient function φ*(n), we investigate ...
Sagar Mandal
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On the sum of Divisors Function
Journal of Number Theory, 1995 A classical identity of Ramanujan states that, for \(\sigma_ z (n)= \sum_{d\mid n} d^ z\) and \(\text{Re } s>1+ \max (0, \text{Re } a\), \(\text{Re } b,\text{Re} (a+b))\) we have \[ \sum_{n=1}^ \infty \sigma_ a (n) \sigma_ b (n) n^{-s}= {{\zeta (s) \zeta (s-a) \zeta (s-b) \zeta (s-a-b)} \over {\zeta (2s-a -b)}}.
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A note on newly introduced arithmetic functions φ+ and σ+ [PDF]
Notes on Number Theory and Discrete MathematicsIn a recent paper [7], the authors introduced new arithmetic functions φ⁺, σ⁺ related to the classical functions φ, and σ, respectively. In this note, we study the behavior of Σ_{n≤x, ω(n)=2}(φ⁺-φ)(n), and Σ_{n≤x, ω(n)=2}(σ⁺-σ)(n), for any real number x ...
Sagar Mandal
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