Results 11 to 20 of about 98,048 (175)
Summation Formulas Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers [PDF]
Abstract and Applied Analysis, 2014 A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics.
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Odd harmonic numbers exceed 10²⁴ [PDF]
Mathematics of Computation, 2010 A numbern>1n>1is harmonic ifσ(n)∣nτ(n)\sigma (n)\mid n\tau (n), whereτ(n)\tau (n)andσ(n)\sigma (n)are the number of positive divisors ofnnand their sum, respectively. It is known that there are no odd harmonic numbers up to101510^{15}. We show here that, for any odd numbern>106n>10^6,τ(n)≤n1/3\tau (n)\le n^{1/3}.
Cohen, Graeme L., Sorli, Ronald M.
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Families of Integrals of Polylogarithmic Functions
Mathematics, 2019 We give an overview of the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider polylogarithmic functions with linear, quadratic, and trigonometric arguments, thereby producing ...
Anthony Sofo
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Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion
Mathematics, 2022 Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers.
Kwang-Wu Chen
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On the denominators of harmonic numbers, III [PDF]
Periodica Mathematica Hungarica, 2022 Let ℒ be the set of all positive integers n such that the denominator of 1+1/2+⋯+1/n is less than the least common multiple of 1,2,⋯,n. In this paper, under a certain assumption on linear independence, we prove that the set ℒ has the upper asymptotic density 1. The assumption follows from Schanuel’s conjecture.
Wu, Bing-Ling, Yan, Xiao-Hui
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rf acceleration with harmonic number jump
Physical Review Special Topics. Accelerators and Beams, 2006 We have recently considered acceleration of protons and heavy ions in a fixed-field alternating-gradient accelerator with nonscaling lattice and linear field profile.
Alessandro G. Ruggiero
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Harmonic numbers, harmonic series and zeta function [PDF]
Moroccan Journal of Pure and Applied Analysis, 2018 AbstractThis paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the seriesis(z)=∑n=1∞μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{
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Certain Hermite–Hadamard Inequalities for Logarithmically Convex Functions with Applications
Mathematics, 2019 In this paper, we discuss various estimates to the right-hand (resp. left-hand) side of the Hermite⁻Hadamard inequality for functions whose absolute values of the second (resp. first) derivatives to positive real powers are log-convex.
Shilpi Jain +3 more
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Some identities involving harmonic numbers [PDF]
Mathematics of Computation, 1990 Let H n {H_n} denote the nth harmonic number. Explicit formulas for sums of the form ∑ a k H k \sum {a_k}{H_k} or ∑ a
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Analysis and application of real-time compensation of positioning precision of the turntable with a harmonic function [PDF]
Metrology and Measurement Systems, 2022 In order to guarantee the accuracy of turntable angle measurement, a real-time compensation method for turntable positioning precision based on harmonic analysis is proposed in this paper.
Yi Zhou +5 more
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