Editing Riemann zeta function (section)

==Applications==
The zeta function occurs in applied [[statistics]] including [[Zipf's law]], [[Zipf–Mandelbrot law]], and [[Lotka's law]].

[[Zeta function regularization]] is used as one possible means of [[regularization (physics)|regularization]] of [[divergent series]] and [[divergent integral]]s in [[quantum field theory]]. In one notable example, the Riemann zeta function shows up explicitly in one method of calculating the [[Casimir effect]]. The zeta function is also useful for the analysis of [[dynamical systems]].<ref>{{cite web|url=http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/spinchains.htm |title=Work on spin-chains by A. Knauf, et. al |website=Empslocal.ex.ac.uk |access-date=2017-01-04}}</ref>

===Musical tuning===
In the theory of [[musical tuning]]s, the zeta function can be used to find [[Equal temperament|equal divisions of the octave]] (EDOs) that closely approximate the intervals of the [[Harmonic series (music)|harmonic series]]. For increasing values of <math>t \in \mathbb{R}</math>, the value of

:<math>\left\vert \zeta \left( \frac{1}{2} + \frac{2\pi{i}}{\ln{(2)}}t \right) \right\vert</math>

peaks near integers that correspond to such EDOs.<ref>{{cite web|url=https://oeis.org/A117536 |title=Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i*2*Pi/log(2)*t)) for increasing real t |author=Gene Ward Smith |website=The On-Line Encyclopedia of Integer Sequences |access-date=2022-03-04}}</ref> Examples include popular choices such as 12, 19, and 53.<ref>{{cite book|title=Tuning, Timbre, Spectrum, Scale |author=William A. Sethares |date=2005 |edition=2nd |publisher=Springer-Verlag London |page=74 |quote=...there are many different ways to evaluate the goodness, reasonableness, fitness, or quality of a scale...Under some measures, 12-tet is the winner, under others 19-tet appears best, 53-tet often appears among the victors...}}</ref>

===Infinite series===
The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.<ref>Most of the formulas in this section are from § 4 of J. M. Borwein et al. (2000)</ref>
*<math>\sum_{n=2}^\infty\bigl(\zeta(n)-1\bigr) = 1</math>

In fact the even and odd terms give the two sums
*<math>\sum_{n=1}^\infty\bigl(\zeta(2n)-1\bigr)=\frac{3}{4}</math>
and

*<math>\sum_{n=1}^\infty\bigl(\zeta(2n+1)-1\bigr)=\frac{1}{4}</math>
Parametrized versions of the above sums are given by

*<math>\sum_{n=1}^\infty(\zeta(2n)-1)\,t^{2n} = \frac{t^2}{t^2-1} + \frac{1}{2} \left(1- \pi t\cot(t\pi)\right)</math>
and

*<math>\sum_{n=1}^\infty(\zeta(2n+1)-1)\,t^{2n} = \frac{t^2}{t^2-1} -\frac{1}{2}\left(\psi^0(t)+\psi^0(-t) \right) - \gamma</math>

with <math>|t|<2</math> and where <math>\psi</math> and <math>\gamma</math> are the [[polygamma function]] and [[Euler's constant]], respectively, as well as

*<math>\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n}\,t^{2n} = \log\left(\dfrac{1-t^2}{\operatorname{sinc}(\pi\,t)}\right)</math>

all of which are continuous at <math>t=1</math>. Other sums include

*<math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n} = 1-\gamma</math>
*<math>\sum_{n=1}^\infty\frac{\zeta(2n)-1}{n} = \ln 2</math>
*<math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n} \left(\left(\tfrac{3}{2}\right)^{n-1}-1\right) = \frac{1}{3} \ln \pi</math>
*<math>\sum_{n=1}^\infty\bigl(\zeta(4n)-1\bigr) = \frac78-\frac{\pi}{4}\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right).</math>
*<math>\sum_{n=2}^\infty\frac{\zeta(n)-1}{n}\Im \bigl((1+i)^n-1-i^n\bigr) = \frac{\pi}{4}</math>

where <math>\Im</math> denotes the [[imaginary part]] of a complex number.

Another interesting series that relates to the [[natural logarithm]] of the [[lemniscate constant]] is the following

*<math>\sum_{n=2}^\infty\left[\frac{2(-1)^n\zeta(n)}{4^n n}-\frac{(-1)^n\zeta(n)}{2^n n} \right]= \ln \left( \frac{\varpi}{2\sqrt2} \right)
</math>

There are yet more formulas in the article [[Harmonic number#Relation to the Riemann zeta function|Harmonic number.]]