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- Harmonic_number abstract "In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:This also equals n times the inverse of the harmonic mean of these natural numbers.Harmonic numbers were studied in antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.The associated harmonic series grows without limit, albeit very slowly, roughly approaching the natural logarithm function. In 1737, Leonhard Euler used the divergence of this series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is the n-th harmonic number. This leads to a variety of surprising conclusions in the Long Tail and the theory of network value.Bertrand's postulate entails that, except for the case n=1, the harmonic numbers are never integers.".
- Harmonic_number thumbnail HarmonicNumbers.svg?width=300.
- Harmonic_number wikiPageExternalLink v11i1n15.pdf.
- Harmonic_number wikiPageExternalLink How%20Euler%20Did%20It%2002%20Estimating%20the%20Basel%20Problem.pdf.
- Harmonic_number wikiPageExternalLink harmonic.pdf.
- Harmonic_number wikiPageExternalLink HarmonicNumberIds.pdf.
- Harmonic_number wikiPageID "214729".
- Harmonic_number wikiPageRevisionID "606042836".
- Harmonic_number hasPhotoCollection Harmonic_number.
- Harmonic_number id "3421".
- Harmonic_number title "Harmonic Number".
- Harmonic_number title "Harmonic number".
- Harmonic_number urlname "HarmonicNumber".
- Harmonic_number subject Category:Number_theory.
- Harmonic_number comment "In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:This also equals n times the inverse of the harmonic mean of these natural numbers.Harmonic numbers were studied in antiquity and are important in various branches of number theory.".
- Harmonic_number label "Harmonic number".
- Harmonic_number label "Harmonisch getal".
- Harmonic_number label "Número armónico".
- Harmonic_number label "調和数 (発散列)".
- Harmonic_number label "調和數".
- Harmonic_number sameAs Número_armónico.
- Harmonic_number sameAs 調和数_(発散列).
- Harmonic_number sameAs Harmonisch_getal.
- Harmonic_number sameAs m.01ffqt.
- Harmonic_number sameAs Q13407133.
- Harmonic_number sameAs Q13407133.
- Harmonic_number wasDerivedFrom Harmonic_number?oldid=606042836.
- Harmonic_number depiction HarmonicNumbers.svg.
- Harmonic_number isPrimaryTopicOf Harmonic_number.