Editing Riemann zeta function (section)

==Specific values==
{{main|Particular values of the Riemann zeta function}}
For any positive even integer {{math|2''n''}},
<math display="block"> \zeta(2n) = \frac{|{B_{2n}}|(2\pi)^{2n}}{2(2n)!},</math>
where {{math|''B''<sub>2''n''</sub>}} is the {{math|2''n''}}-th [[Bernoulli number]].
For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic {{mvar|K}}-theory of the integers; see [[Special values of L-functions|Special values of {{mvar|L}}-functions]].

For nonpositive integers, one has
<math display="block">\zeta(-n)= -\frac{B_{n+1}}{n+1}</math>
for {{math|''n'' ≥ 0}} (using the convention that {{math|''B''<sub>1</sub> {{=}} {{sfrac|1|2}}}}).
In particular, {{mvar|ζ}} vanishes at the negative even integers because {{math|''B''<sub>''m''</sub> {{=}} 0}} for all odd {{mvar|m}} other than&nbsp;1. These are the so-called "trivial zeros" of the zeta function.

Via [[analytic continuation]], one can show that
<math display="block">\zeta(-1) = -\tfrac{1}{12}</math>
This gives a pretext for assigning a finite value to the divergent series [[1 + 2 + 3 + 4 + ⋯]], which has been used in certain contexts ([[Ramanujan summation]]) such as [[string theory]].<ref name='polchinski'>{{cite book |last=Polchinski |first=Joseph |author-link=Joseph Polchinski |series=String Theory |volume=I |title=An Introduction to the Bosonic String |publisher=Cambridge University Press |year=1998 |page=22 |isbn=978-0-521-63303-1}}</ref>  Analogously, the particular value
<math display="block">\zeta(0) = -\tfrac{1}{2}</math>
can be viewed as assigning a finite result to the divergent series [[1 + 1 + 1 + 1 + ⋯]].

The value
<math display="block">\zeta\bigl(\tfrac12\bigr) = -1.46035450880958681288\ldots</math>
is employed in calculating kinetic boundary layer problems of linear kinetic equations.<ref>{{cite journal|first1=A. J. |last1=Kainz |first2=U. M. |last2=Titulaer |title=An accurate two-stream moment method for kinetic boundary layer problems of linear kinetic equations |pages=1855–1874 |journal=J. Phys. A: Math. Gen. |volume=25 |issue=7 |date=1992|bibcode=1992JPhA...25.1855K |doi=10.1088/0305-4470/25/7/026 }}</ref><ref>Further digits and references for this constant are available at {{OEIS2C|id=A059750}}.</ref>

Although
<math display="block">\zeta(1) = 1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots</math>
diverges, its [[Cauchy principal value]]
<math display="block"> \lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}</math>
exists and is equal to the [[Euler–Mascheroni constant]] {{math|''γ'' {{=}} 0.5772...}}.<ref name=Sondow1998>{{cite journal |last1=Sondow |first1=Jonathan |date=1998 |title=An antisymmetric formula for Euler's constant |journal=Mathematics Magazine |volume=71 |issue=3 |pages=219–220 |doi=10.1080/0025570X.1998.11996638 |access-date=2006-05-29 |url=http://home.earthlink.net/~jsondow/id8.html |archive-date=2011-06-04 |archive-url=https://web.archive.org/web/20110604123534/http://home.earthlink.net/~jsondow/id8.html}}</ref>

The demonstration of the particular value
<math display="block">\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}</math>
is known as the [[Basel problem]]. The reciprocal of this sum answers the question: ''What is the probability that two numbers selected at random are [[coprime|relatively prime]]?''<ref>{{cite book|author-link=C. Stanley Ogilvy|first1=C. S. |last1=Ogilvy |first2=J. T. |last2=Anderson |title=Excursions in Number Theory |pages=29–35 |publisher=Dover Publications |date=1988 |isbn=0-486-25778-9}}</ref>
The value
<math display="block">\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.202056903159594285399...</math>
is [[Apéry's constant]].

Taking the limit <math>s \rightarrow +\infty</math> through the real numbers, one obtains <math>\zeta (+\infty) = 1</math>. But at [[complex infinity]] on the [[Riemann sphere]] the zeta function has an [[essential singularity]].<ref name=":0">{{Cite journal|last1=Steuding|first1=Jörn|last2=Suriajaya|first2=Ade Irma|date=2020-11-01|title=Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines|journal=Computational Methods and Function Theory|language=en|volume=20|issue=3|pages=389–401|doi=10.1007/s40315-020-00316-x|s2cid=216323223 |issn=2195-3724|quote=Theorem 2 implies that ζ has an essential singularity at infinity|doi-access=free|hdl=2324/4483207|hdl-access=free}}</ref>