en>Gareth Jones: /* One dimension */ coauthors

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One dimension: coauthors

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In mathematics, a '''Shimizu L-function''', introduced by {{harvs|txt|authorlink=Hideo Shimizu|year=1963|last=Shimizu}} is a [[Dirichlet series]] associated to a [[totally real number field|totally real]] [[algebraic number field]].
{{harvs|txt | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Donnelly | first2=H. | last3=Singer | first3=I. M. | author3-link=Isadore Singer | title=Eta invariants, signature defects of cusps, and values of L-functions | url=http://dx.doi.org/10.2307/2006957 | doi=10.2307/2006957 | id={{MR|707164}} | year=1983 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=118 | issue=1 | pages=131–177}}
defined the [[signature defect]] of the boundary of a manifold as the [[eta invariant]], the value as ''s''=0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a [[Hilbert modular surface]] can be expressed in terms of the value at ''s''=0 or 1 of a Shimizu L-function.

==Definition==

Suppose that ''K'' is a totally real algebraic number field, ''M'' is a lattice in the field, and ''V'' is a subgroup of maximal rank of the group of totally positive units preserving the lattice. The Shimazu L-series is given by
:<math>L(M,V,s) = \sum_{\mu\in \{M-0\}/V} \frac{\operatorname{sign} N(\mu)}{|N(\mu)|^s}</math>

==References==

*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Donnelly | first2=H. | last3=Singer | first3=I. M. | title=Geometry and analysis of Shimizu L-functions | url=http://www.jstor.org/stable/12685 | id={{MR|674920}} | year=1982 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=79 | issue=18 | pages=5751}}
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Donnelly | first2=H. | last3=Singer | first3=I. M. | title=Eta invariants, signature defects of cusps, and values of L-functions | url=http://dx.doi.org/10.2307/2006957 | doi=10.2307/2006957 | id={{MR|707164}} | year=1983 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=118 | issue=1 | pages=131–177}}
*{{Citation | last1=Shimizu | first1=Hideo | title=On discontinuous groups operating on the product of the upper half planes | url=http://www.jstor.org/stable/1970201 | id={{MR|0145106}} | year=1963 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=77 | pages=33–71}}

[[Category:Zeta and L-functions]]

Reflected Brownian motion - Revision history

en>Gareth Jones: /* One dimension */ coauthors