Editing Riemann zeta function (section)

==Generalizations==<!-- This section is linked from [[Power law]] -->
There are a number of related [[zeta function]]s that can be considered to be generalizations of the Riemann zeta function. These include the [[Hurwitz zeta function]]

:<math>\zeta(s,q) = \sum_{k=0}^\infty \frac{1}{(k+q)^s}</math>
(the convergent series representation was given by [[Helmut Hasse]] in 1930,<ref name = Hasse1930>{{Cite journal |first=Helmut |last=Hasse |author-link=Helmut Hasse |title=Ein Summierungsverfahren für die Riemannsche {{mvar|ζ}}-Reihe |trans-title=A summation method for the Riemann ζ series |year=1930 |journal=[[Mathematische Zeitschrift]] |volume=32 |issue=1 |pages=458–464 |doi=10.1007/BF01194645 |s2cid=120392534 |language=de}}</ref> cf. [[Hurwitz zeta function]]), which coincides with the Riemann zeta function when {{math|''q'' {{=}} 1}} (the lower limit of summation in the Hurwitz zeta function is 0, not 1), the [[Dirichlet L-function|Dirichlet {{mvar|L}}-functions]] and the [[Dedekind zeta function]]. For other related functions see the articles [[zeta function]] and [[L-function|{{mvar|L}}-function]].

The [[polylogarithm]] is given by

:<math>\operatorname{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}</math>

which coincides with the Riemann zeta function when {{math|''z'' {{=}} 1}}.
The [[Clausen function]] {{math|Cl<sub>''s''</sub>(''θ'')}} can be chosen as the real or imaginary part of {{math|Li<sub>''s''</sub>(''e''{{isup|''iθ''}})}}.

The [[Lerch transcendent]] is given by
:<math>\Phi(z, s, q) = \sum_{k=0}^\infty\frac { z^k} {(k+q)^s}</math>
which coincides with the Riemann zeta function when {{math|''z'' {{=}} 1}} and {{math|''q'' {{=}} 1}} (the lower limit of summation in the Lerch transcendent is&nbsp;0, not&nbsp;1).

The [[multiple zeta functions]] are defined by

:<math>\zeta(s_1,s_2,\ldots,s_n) = \sum_{k_1>k_2>\cdots>k_n>0} {k_1}^{-s_1}{k_2}^{-s_2}\cdots {k_n}^{-s_n}.</math>

One can analytically continue these functions to the {{mvar|n}}-dimensional complex space. The special values taken by these functions at positive integer arguments are called [[multiple zeta values]] by number theorists and have been connected to many different branches in mathematics and physics.