Results 11 to 20 of about 10,640,447 (355)
Harmonic sets and the harmonic prime number theorem [PDF]
Bulletin of the Australian Mathematical Society, 2005 We restrict primes and prime powers to sets H(x)= U∞n=1 (x/2n, x/(2n-1)). Let θH(x)= ∑ pεH(x)log p. Then the error in θH(x) has, unconditionally, the expected order of magnitude θH (x)= xlog2 + O(√x). However, if ψH(x)= ∑pmε H(x) log p then ψH(x)= xlog2+
Broughan, Kevin A., Casey, Rory J.
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Ramanujan's Harmonic Number Expansion [PDF]
, 2005 7 pages. Acknowledgements added. Many typos corrected.
Mark B. Villarino
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Unitary Harmonic Numbers [PDF]
Proceedings of the American Mathematical Society, 1975 If d ∗ ( n ) {d^ \ast }(n) and σ ∗ ( n ) {\sigma ^ \ast }(n) denote the number and sum, respectively, of the unitary divisors of the natural number n n
Hagis, Peter jun., Lord, Graham
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Proceedings - Mathematical Sciences, 2021 The harmonic sums \(\sum_{r=k+1}^n \frac{1}{r}\) are not integers for any \(k \geq 1\). One way of proving this uses the Bertrand postulate that there is always a prime strictly between \(k\) and \(2k\) if \(k \geq 2\). The paper under review considers the more general analogous sums \(h_K(n) := \sum_{r=1}^n \frac{a_r}{r}\) where \(a_r\) is the number ...
Çağatay Altuntaş, Haydar Göral
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A note on harmonic number identities, Stirling series and multiple zeta values [PDF]
International Journal of Number Theory, 2018 We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa–Kaneko zeta function, studied before amongst others by Candelpergher and Coppo ...
Markus Kuba, A. Panholzer
semanticscholar +1 more source
Infinitary harmonic numbers [PDF]
Bulletin of the Australian Mathematical Society, 1990 The infinitary divisors of a natural number n are the products of its divisors of the , where py is an exact prime-power divisor of n and (where yα = 0 or 1) is the binary representation of y. Infinitary harmonic numbers are those for which the infinitary divisors have integer harmonic mean.
Hagis, Peter jun., Cohen, Graeme L.
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Harmonic numbers and finite groups [PDF]
Rendiconti del Seminario Matematico della Università di Padova, 2014 Given a finite group G , let {\tau} (G) be the number of normal subgroups of G ...
Baishya, Sekhar Jyoti, Das, Ashish Kumar
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Generalization in diffusion models arises from geometry-adaptive harmonic representation [PDF]
International Conference on Learning Representations, 2023 Deep neural networks (DNNs) trained for image denoising are able to generate high-quality samples with score-based reverse diffusion algorithms. These impressive capabilities seem to imply an escape from the curse of dimensionality, but recent reports of
Zahra Kadkhodaie +3 more
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Umbral Methods and Harmonic Numbers
Axioms, 2018 The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.
Dattoli G. +3 more
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