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| In [[mathematics]], '''Dedekind sums''', named after [[Richard Dedekind]], are certain sums of products of a [[sawtooth function]], and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the [[functional equation]] of the [[Dedekind eta function]]. They have subsequently been much studied in [[number theory]], and have occurred in some problems of [[topology]]. Dedekind sums obey a large number of relationships on themselves; this article lists only a tiny fraction of these.
| | 47 year-old Marine Biologist Harry Crosser from Drumheller, loves to spend time painting, property developers in singapore and tesla coils. Did a cruise ship experience that was comprised of passing by The Sassi and the Park of the Rupestrian Churches of Matera.<br><br>my web blog; [http://computerlessons.pro/node/71045 computerlessons.pro] |
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| == Definition ==
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| Define the [[sawtooth function]] <math>\left( \left( \right) \right):\mathbb{R} \rightarrow \mathbb{R}</math> as
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| :<math>((x))=\begin{cases}
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| x-\lfloor x\rfloor - 1/2, &\mbox{if }x\in\mathbb{R}\setminus\mathbb{Z};\\
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| 0,&\mbox{if }x\in\mathbb{Z}.
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| \end{cases}</math>
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| We then let
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| :''D'' :'''Z'''<sup>3</sup> → '''R'''
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| be defined by
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| :<math>D(a,b;c)=\sum_{n \bmod c} \left( \Bigg( \frac{an}{c} \Bigg) \right) \left( \left( \frac{bn}{c} \right) \right),</math>
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| the terms on the right being the '''Dedekind sums'''. For the case ''a''=1, one often writes
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| :''s''(''b'',''c'') = ''D''(1,''b'';''c'').
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| ==Simple formulae==
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| Note that ''D'' is symmetric in ''a'' and ''b'', and hence
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| :<math>D(a,b;c)=D(b,a;c),</math>
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| and that, by the oddness of (()), | |
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| :''D''(−''a'',''b'';''c'') = −''D''(''a'',''b'';''c''),
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| :''D''(''a'',''b'';−''c'') = ''D''(''a'',''b'';''c'').
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| By the periodicity of ''D'' in its first two arguments, the third argument being the length of the period for both,
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| :''D''(''a'',''b'';''c'')=''D''(''a''+''kc'',''b''+''lc'';''c''), for all integers ''k'',''l''.
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| If ''d'' is a positive integer, then
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| :''D''(''ad'',''bd'';''cd'') = ''dD''(''a'',''b'';''c''),
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| :''D''(''ad'',''bd'';''c'') = ''D''(''a'',''b'';''c''), if (''d'',''c'') = 1,
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| :''D''(''ad'',''b'';''cd'') = ''D''(''a'',''b'';''c''), if (''d'',''b'') = 1.
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| There is a proof for the last equality making use of
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| :<math>\sum_{n \bmod c} \left( \left( \frac{n+x}{c} \right) \right)=\left(\left( x\right)\right),\qquad\forall x\in\mathbb{R}.</math>
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| Furthermore, ''az'' = 1 (mod ''c'') implies ''D''(''a'',''b'';''c'') = ''D''(1,''bz'';''c'').
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| ==Alternative forms==
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| If ''b'' and ''c'' are coprime, we may write ''s''(''b'',''c'') as
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| :<math>s(b,c)=\frac{-1}{c} \sum_\omega
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| \frac{1} { (1-\omega^b) (1-\omega ) }
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| +\frac{1}{4} - \frac{1}{4c},</math>
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| where the sum extends over the ''c''-th roots of unity other than 1, i.e. over all <math>\omega</math> such that <math>\omega^c=1</math> and <math>\omega\not=1</math>.
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| If ''b'', ''c'' > 0 are coprime, then
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| :<math>s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1}
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| \cot \left( \frac{\pi n}{c} \right)
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| \cot \left( \frac{\pi nb}{c} \right).
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| </math>
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| ==Reciprocity law==
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| If ''b'' and ''c'' are coprime positive integers then
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| :<math>s(b,c)+s(c,b) =\frac{1}{12}\left(\frac{b}{c}+\frac{1}{bc}+\frac{c}{b}\right)-\frac{1}{4}.</math>
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| Rewriting this as
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| :<math>12bc \left( s(b,c) + s(c,b) \right) = b^2 + c^2 -3bc + 1,</math>
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| it follows that the number 6''c'' ''s''(''b'',''c'') is an integer.
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| If ''k'' = (3, ''c'') then
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| :<math> 12bc\, s(c,b)=0 \mod kc</math>
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| and
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| :<math> 12bc\, s(b,c)=b^2+1 \mod kc.</math>
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| A relation that is prominent in the theory of the [[Dedekind eta function]] is the following. Let ''q'' = 3, 5, 7 or 13 and let ''n'' = 24/(''q'' − 1). Then given integers ''a'', ''b'', ''c'', ''d'' with ''ad'' − ''bc'' = 1 (thus belonging to the [[modular group]]), with ''c'' chosen so that ''c'' = ''kq'' for some integer ''k'' > 0, define
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| :<math>\delta = s(a,c) - \frac{a+d}{12c} - s(a,k) + \frac{a+d}{12k}</math>
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| Then one has ''n''δ is an even integer.
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| ==Rademacher's generalization of the reciprocity law==
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| [[Hans Rademacher]] found the following generalization of the reciprocity law for Dedekind sums:<ref>{{cite journal | last=Rademacher | first=Hans | authorlink=Hans Rademacher | title=Generalization of the reciprocity formula for Dedekind sums | journal=[[Duke Mathematical Journal]] | volume=21 | pages=391-397 | year=1954 | zbl=0057.03801 }}</ref> If ''a'',''b'', and ''c'' are pairwise coprime positive integers, then
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| :<math>D(a,b;c)+D(b,c;a)+D(c,a;b)=\frac{1}{12}\frac{a^2+b^2+c^2}{abc}-\frac{1}{4}.</math>
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| ==References==
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| <references/>
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| * [[Tom M. Apostol]], ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 ''(See chapter 3.)''
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| * Matthias Beck and Sinai Robins, ''[http://math.sfsu.edu/beck/papers/dedekind.slides.pdf Dedekind sums: a discrete geometric viewpoint]'', (2005 or earlier)
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| * [[Hans Rademacher]] and [[Emil Grosswald]], ''Dedekind Sums'', Carus Math. Monographs, 1972. ISBN 0-88385-016-8.
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| [[Category:Number theory]]
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| [[Category:Modular forms]]
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47 year-old Marine Biologist Harry Crosser from Drumheller, loves to spend time painting, property developers in singapore and tesla coils. Did a cruise ship experience that was comprised of passing by The Sassi and the Park of the Rupestrian Churches of Matera.
my web blog; computerlessons.pro