https://en.formulasearchengine.com/index.php?action=history&feed=atom&title=Ruelle_zeta_function Ruelle zeta function - Revision history 2026-04-17T13:55:26Z Revision history for this page on the wiki MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=Ruelle_zeta_function&diff=28142&oldid=prev en>David Eppstein: Avoid unnecessary pipe in link. Remove technical tag; it is technical, but I think unavoidably so. 2013-12-26T20:44:05Z <p>Avoid unnecessary pipe in link. Remove technical tag; it is technical, but I think unavoidably so.</p> <p><b>New page</b></p><div>In mathematics, &#039;&#039;&#039;octic reciprocity&#039;&#039;&#039; is a [[reciprocity law]] relating the residues of 8th powers [[modular arithmetic|modulo]] [[prime number|primes]], analogous to the [[law of quadratic reciprocity]].<br /> <br /> There is a [[rational reciprocity law]] for 8th powers, due to Williams. Define the symbol (&#039;&#039;x&#039;&#039;|&#039;&#039;p&#039;&#039;)&lt;sub&gt;&#039;&#039;k&#039;&#039;&lt;/sub&gt; to be +1 if &#039;&#039;x&#039;&#039; is a &#039;&#039;k&#039;&#039;-th power modulo the prime &#039;&#039;p&#039;&#039; and -1 otherwise. Let &#039;&#039;p&#039;&#039; and &#039;&#039;q&#039;&#039; be distinct primes congruent to 1 modulo 8, such that (&#039;&#039;p&#039;&#039;|&#039;&#039;q&#039;&#039;) = (&#039;&#039;q&#039;&#039;|&#039;&#039;p&#039;&#039;) = +1. Let &#039;&#039;p&#039;&#039; = &#039;&#039;a&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; + &#039;&#039;b&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; = &#039;&#039;c&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; + 2&#039;&#039;d&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; and &#039;&#039;q&#039;&#039; = &#039;&#039;A&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; + &#039;&#039;B&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; = &#039;&#039;C&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; + 2&#039;&#039;D&#039;&#039;&lt;sup&gt;2&lt;/sup&gt;, with &#039;&#039;aA&#039;&#039; odd. Then<br /> <br /> :&lt;math&gt; (p|q)_8 = (q|p)_8 = (aB-bA|q)_4 (cD-dC|q)_2 \ .&lt;/math&gt;<br /> <br /> ==References==<br /> {{reflist}}<br /> *{{citation|mr=1761696|zbl=0949.11002 | last=Lemmermeyer|first= Franz<br /> |title=Reciprocity laws. From Euler to Eisenstein|series= Springer Monographs in Mathematics|publisher= Springer-Verlag, Berlin|year= 2000|isbn= 3-540-66957-4 |url=http://books.google.com/books?id=EwjpPeK6GpEC | pages=289-316 }}<br /> *{{Citation | last1=Williams | first1=Kenneth S. | title=A rational octic reciprocity law | url= http://projecteuclid.org/euclid.pjm/1102867415 | mr=0414467 | year=1976 | journal=[[Pacific Journal of Mathematics]] | issn=0030-8730 | volume=63 | issue=2 | pages=563–570 | zbl=0311.10004 }}<br /> <br /> [[Category:Algebraic number theory]]<br /> <br /> {{numtheory-stub}}</div> en>David Eppstein