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2014-07-23T07:25:47Z

Avoiding dangling pointer errors

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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]

	~~== Definition ==~~
	~~Define the [[sawtooth function]] <math>\left( \left( \right) \right):\mathbb{R} \rightarrow \mathbb{R}</math> as~~
	~~:<math>((x))=\begin{cases}~~
	~~x-\lfloor x\rfloor - 1/2, &\mbox{if }x\in\mathbb{R}\setminus\mathbb{Z};\\~~
	~~0,&\mbox{if }x\in\mathbb{Z}.~~
	~~\end{cases}</math>~~

	~~We then let~~

	~~:''D'' :'''Z'''<sup>3</sup> → '''R'''~~

	~~be defined by~~

	~~:<math>D(a,b;c)=\sum_{n \bmod c} \left( \Bigg( \frac{an}{c} \Bigg) \right) \left( \left( \frac{bn}{c} \right) \right),</math>~~

	~~the terms on the right being the '''Dedekind sums'''. For the case ''a''=1, one often writes~~

	~~:''s''(''b'',''c'') = ''D''(1,''b'';''c'').~~

	~~==Simple formulae==~~
	~~Note that ''D'' is symmetric in ''a'' and ''b'', and hence~~

	~~:<math>D(a,b;c)=D(b,a;c),</math>~~

	and ~~that, by the oddness of (()),~~

	~~:''D''(−''a'',''b'';''c'') = −''D''(''a'',''b'';''c''),~~

	~~:''D''(''a'',''b'';−''c'') = ''D''(''a'',''b'';''c'')~~.

	~~By the periodicity of ''D'' in its first two arguments, the third argument being the length of the period for both,~~

	~~:''D''(''a'',''b'';''c'')=''D''(''~~a~~''+''kc'',''b''+''lc'';''c''), for all integers ''k'',''l''.~~

	~~If ''d'' is a positive integer, then~~

	~~:''D''(''ad'',''bd'';''cd'') = ''dD''(''a'',''b'';''c''),~~

	~~:''D''(''ad'',''bd'';''c'') = ''D''(''a'',''b'';''c''), if (''d'',''c'') = 1,~~

	~~:''D''(''ad'',''b'';''cd'') = ''D''(''a'',''b'';''c''), if (''d'',''b'') = 1.~~

	~~There is a proof for the last equality making use of~~

	~~:<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>~~

	~~Furthermore, ''az'' = 1 (mod ''c'') implies ''D''(''a'',''b'';''c'') = ''D''(1,''bz'';''c'').~~

	~~==Alternative forms==~~

	~~If ''b'' and ''c'' are coprime, we may write ''s''(''b'',''c'') as~~

	~~:<math>s(b,c)=\frac{-1}{c} \sum_\omega~~
	~~\frac{1} { (1-\omega^b) (1-\omega ) }~~
	~~+\frac{1}{4} - \frac{1}{4c},</math>~~

	~~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>.~~

	~~If ''b'', ''c'' > 0 are coprime, then~~

	~~:<math>s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1}~~
	~~\cot \left( \frac{\pi n}{c} \right)~~
	~~\cot \left( \frac{\pi nb}{c} \right).~~
	~~</math>~~

	~~==Reciprocity law==~~
	~~If ''b'' and ''c'' are coprime positive integers then~~

	~~:<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>~~

	~~Rewriting this as~~

	~~:<math>12bc \left( s(b,c) + s(c,b) \right) = b^2 + c^2 -3bc + 1,</math>~~

	~~it follows that the number 6''c'' ''s''(''b'',''c'') is an integer.~~

	~~If ''k'' = (3, ''c'') then~~

	~~:<math> 12bc\, s(c,b)=0 \mod kc</math>~~

	~~and~~

	~~:<math> 12bc\, s(b,c)=b^2+1 \mod kc.</math>~~

	~~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~~

	~~:<math>\delta = s(a,c) - \frac{a+d}{12c} - s(a,k) + \frac{a+d}{12k}</math>~~

	~~Then one has ''n''δ is an even integer.~~

	~~==Rademacher's generalization~~ of the ~~reciprocity law==~~
	~~[[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~~

	:<~~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>~~

	~~==References==~~
	~~<references/>~~
	* [[Tom M. Apostol]], ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 ''(See chapter 3.)''
	* Matthias Beck and Sinai Robins, ''[http://~~math~~.~~sfsu.edu/beck~~/~~papers~~/~~dedekind.slides.pdf Dedekind sums: a discrete geometric viewpoint]'', (2005 or earlier)~~
	* [[Hans Rademacher]] and [[Emil Grosswald]], ''Dedekind Sums'', Carus Math. Monographs, 1972. ~~ISBN 0-88385-016-8.~~

	~~[[Category:Number theory]]~~
	~~[[Category:Modular forms]~~]

en>Loadmaster: /* Other uses */ New sect. Is there a template for Jargon file terms?

2014-01-07T18:59:07Z

Other uses: New sect. Is there a template for Jargon file terms?

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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.

== Definition ==
Define the [[sawtooth function]] <math>\left( \left( \right) \right):\mathbb{R} \rightarrow \mathbb{R}</math> as
:<math>((x))=\begin{cases}
x-\lfloor x\rfloor - 1/2, &\mbox{if }x\in\mathbb{R}\setminus\mathbb{Z};\\
0,&\mbox{if }x\in\mathbb{Z}.
\end{cases}</math>

We then let

:''D'' :'''Z'''<sup>3</sup> → '''R'''

be defined by

:<math>D(a,b;c)=\sum_{n \bmod c} \left( \Bigg( \frac{an}{c} \Bigg) \right) \left( \left( \frac{bn}{c} \right) \right),</math>

the terms on the right being the '''Dedekind sums'''. For the case ''a''=1, one often writes

:''s''(''b'',''c'') = ''D''(1,''b'';''c'').

==Simple formulae==
Note that ''D'' is symmetric in ''a'' and ''b'', and hence

:<math>D(a,b;c)=D(b,a;c),</math>

and that, by the oddness of (()),

:''D''(−''a'',''b'';''c'') = −''D''(''a'',''b'';''c''),

:''D''(''a'',''b'';−''c'') = ''D''(''a'',''b'';''c'').

By the periodicity of ''D'' in its first two arguments, the third argument being the length of the period for both,

:''D''(''a'',''b'';''c'')=''D''(''a''+''kc'',''b''+''lc'';''c''), for all integers ''k'',''l''.

If ''d'' is a positive integer, then

:''D''(''ad'',''bd'';''cd'') = ''dD''(''a'',''b'';''c''),

:''D''(''ad'',''bd'';''c'') = ''D''(''a'',''b'';''c''), if (''d'',''c'') = 1,

:''D''(''ad'',''b'';''cd'') = ''D''(''a'',''b'';''c''), if (''d'',''b'') = 1.

There is a proof for the last equality making use of

:<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>

Furthermore, ''az'' = 1 (mod ''c'') implies ''D''(''a'',''b'';''c'') = ''D''(1,''bz'';''c'').

==Alternative forms==

If ''b'' and ''c'' are coprime, we may write ''s''(''b'',''c'') as

:<math>s(b,c)=\frac{-1}{c} \sum_\omega
\frac{1} { (1-\omega^b) (1-\omega ) }
+\frac{1}{4} - \frac{1}{4c},</math>

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>.

If ''b'', ''c'' > 0 are coprime, then

:<math>s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1}
\cot \left( \frac{\pi n}{c} \right)
\cot \left( \frac{\pi nb}{c} \right).
</math>

==Reciprocity law==
If ''b'' and ''c'' are coprime positive integers then

:<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>

Rewriting this as

:<math>12bc \left( s(b,c) + s(c,b) \right) = b^2 + c^2 -3bc + 1,</math>

it follows that the number 6''c'' ''s''(''b'',''c'') is an integer.

If ''k'' = (3, ''c'') then

:<math> 12bc\, s(c,b)=0 \mod kc</math>

and

:<math> 12bc\, s(b,c)=b^2+1 \mod kc.</math>

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

:<math>\delta = s(a,c) - \frac{a+d}{12c} - s(a,k) + \frac{a+d}{12k}</math>

Then one has ''n''δ is an even integer.

==Rademacher's generalization of the reciprocity law==
[[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

:<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>

==References==
<references/>
* [[Tom M. Apostol]], ''Modular functions and Dirichlet Series in Number Theory'' (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 ''(See chapter 3.)''
* Matthias Beck and Sinai Robins, ''[http://math.sfsu.edu/beck/papers/dedekind.slides.pdf Dedekind sums: a discrete geometric viewpoint]'', (2005 or earlier)
* [[Hans Rademacher]] and [[Emil Grosswald]], ''Dedekind Sums'', Carus Math. Monographs, 1972. ISBN 0-88385-016-8.

[[Category:Number theory]]
[[Category:Modular forms]]