Riemann zeta function: Difference between revisions
m Removing todo:increase applicability and todo:simplify — this is wishful thinking: it *is* an advanced math topic and its applicability is already presented, namely through zeta edo lists and optimal octave stretch |
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{{Wikipedia|Riemann zeta function}} | {{Wikipedia|Riemann zeta function}} | ||
The Riemann zeta function is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size. | The '''Riemann zeta function''' is a famous mathematical function, best known for its relationship with the Riemann hypothesis, a 200-year old unsolved problem involving the distribution of the prime numbers. However, it also has an intriguing musical interpretation: the zeta function shows how "well" a given [[equal temperament]] approximates the no-limit [[just intonation]] relative to its size. | ||
As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function. | As a result, although the zeta function is best known for its use in analytic number theory, the zeta function is present in the background of some tuning theory—the [[harmonic entropy]] model of [[concordance]] can be shown to be related to the Fourier transform of the zeta function, and several tuning-theoretic metrics, if extended to the infinite-limit, yield expressions that are related to the zeta function. Sometimes these are in terms of the "prime zeta function", which is closely related and can also be derived as a simple expression of the zeta function. | ||
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[[Category:Number theory]] | [[Category:Number theory]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||