இராமானுசன் கூட்டு
:''இக்கட்டுரை [[இராமானுசன் கூட்டுகை]] (Ramanujan summation) என்பது பற்றியது அன்று''
 
 கணிதத்தின் ஒரு பிரிவான எண்கோட்பாட்டியலில், '''இராமானுசன் கூட்டு''' (Ramanujan's sum), என்பதைப் பொதுவாக ''c''&lt;sub&gt;''q''&lt;/sub&gt;(''n''), எனக்குறிப்பது வழக்கம். இது நேர்ம எண் மாறிகள் ''q'', ''n'' ஆகியவற்றால் ஆன சார்பியம் (சார்பு). இதனைக் கீழ்க்காணும் சூத்திரத்தால் குறிக்கலாம்
 
 :&lt;math&gt;c_q(n)=
 \sum_{a=1\atop (a,q)=1}^q
 e^{2 \pi i \tfrac{a}{q} n}
 ,
 &lt;/math&gt;
 
 மேலுள்ளதில் (''a'', ''q'') = 1 என்னும் குறியீடு என்ன குறிக்கின்றதென்றால்,  ''a'' என்பது ''q'' என்னும் எண்ணோடு ஒப்பீட்டு பகா எண்ணின் (co-prime) மதிப்புகளை மட்டுமே கொள்ளும் என்று பொருள்.
 
 [[இராமானுசன்|சீனிவாச இராமானுசன்]] இந்த கூட்டு வாய்பாட்டை 1918 ஆய்வுத்தாளில் அளித்தார்&lt;ref&gt;Ramanujan, ''On Certain Trigonometric Sums ...'' &lt;blockquote&gt;These sums are obviously of great interest, and a few of their properties have been discussed already. But, so far as I know, they have never been considered from the point of view which I adopt in  this paper; and I believe that all the results which it contains are new.&lt;/blockquote&gt;(''Papers'', p. 179). In a footnote cites pp. 360&amp;ndash;370 of the Dirichlet-Dedekind ''Vorlesungen über Zahlentheorie'', 4th ed.&lt;/ref&gt; இக் கூட்டு வாய்பாட்டினை வினோகிராடோவ் தேற்றத்தை(Vinogradov's theorem)நிறுவுவதில் பயன்படுத்தியுள்ளார்கள். இத்தேற்றம் மிகப்பெரிய ஒற்றைப்படை எண்கள் ஒவ்வொன்றும் மூன்று பகா எண்களின் கூட்டுத்தொகை என கூறுகின்றது&lt;ref&gt;Nathanson, ch. 8&lt;/ref&gt;
 ==இராமானுசன் கூட்டு அட்டவணை==
 
 {|  class=&quot;wikitable&quot; style=&quot;text-align:center&quot; cellpadding=&quot;2&quot;
 |+ Ramanujan Sum ''c''&lt;sub&gt;''s''&lt;/sub&gt;(''n'')
 |-
 | colspan=&quot;2&quot; rowspan=&quot;2&quot; | &amp;nbsp;
 ! colspan=&quot;30&quot; | n
 |-
 ! width=&quot;25&quot; | 1
 ! width=&quot;25&quot; | 2
 ! width=&quot;25&quot; | 3
 ! width=&quot;25&quot; | 4
 ! width=&quot;25&quot; | 5
 ! width=&quot;25&quot; | 6
 ! width=&quot;25&quot; | 7
 ! width=&quot;25&quot; | 8 
 ! width=&quot;25&quot; | 9 
 ! width=&quot;25&quot; | 10
 ! width=&quot;25&quot; | 11
 ! width=&quot;25&quot; | 12
 ! width=&quot;25&quot; | 13
 ! width=&quot;25&quot; | 14
 ! width=&quot;25&quot; | 15
 ! width=&quot;25&quot; | 16
 ! width=&quot;25&quot; | 17
 ! width=&quot;25&quot; | 18 
 ! width=&quot;25&quot; | 19 
 ! width=&quot;25&quot; | 20
 ! width=&quot;25&quot; | 21
 ! width=&quot;25&quot; | 22
 ! width=&quot;25&quot; | 23
 ! width=&quot;25&quot; | 24
 ! width=&quot;25&quot; | 25
 ! width=&quot;25&quot; | 26
 ! width=&quot;25&quot; | 27
 ! width=&quot;25&quot; | 28 
 ! width=&quot;25&quot; | 29 
 ! width=&quot;25&quot; | 30
 |-
 ! rowspan=&quot;30&quot; width=&quot;25&quot; | s
 ! width=&quot;25&quot; | 1
 | style=&quot;background:#ffdead;&quot; | 1 
 | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
 |-
 ! 2 
 | &amp;minus;1 
 | style=&quot;background:#ffdead;&quot; | 1 
 | &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1
 | &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1
 | &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1
 |-
 ! 3 
 | &amp;minus;1 || &amp;minus;1 
 | style=&quot;background:#ffdead;&quot; | 2 
 | &amp;minus;1 || &amp;minus;1 || 2 
 | &amp;minus;1 || &amp;minus;1 || 2 || &amp;minus;1 || &amp;minus;1 || 2 
 | &amp;minus;1 || &amp;minus;1 || 2 || &amp;minus;1 || &amp;minus;1 || 2 
 | &amp;minus;1 || &amp;minus;1 || 2 || &amp;minus;1 || &amp;minus;1 || 2 
 | &amp;minus;1 || &amp;minus;1 || 2 || &amp;minus;1 || &amp;minus;1 || 2 
 |-
 ! 4 
 | 0 || &amp;minus;2 || 0 
 | style=&quot;background:#ffdead;&quot; | 2 
 | 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 
 | 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 
 | 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 
 |-
 ! 5 
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || style=&quot;background:#ffdead;&quot; | 4
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 4
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 4
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 4
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 4
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 4
 |-
 ! 6 
 | 1 || &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || 1 || style=&quot;background:#ffdead;&quot; | 2 
 | 1 || &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || 1 || 2 
 | 1 || &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || 1 || 2 
 | 1 || &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || 1 || 2 
 | 1 || &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || 1 || 2 
 |-
 ! 7 
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || style=&quot;background:#ffdead;&quot; | 6 
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 6 
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 6 
 | &amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 ||&amp;minus;1 || 6 
 | &amp;minus;1 ||&amp;minus;1 
 |-
 ! 8 
 | 0 || 0 || 0 || &amp;minus;4 || 0 || 0 || 0 || style=&quot;background:#ffdead;&quot; | 4  
 | 0 || 0 || 0 || &amp;minus;4 || 0 || 0 || 0 || 4
 | 0 || 0 || 0 || &amp;minus;4 || 0 || 0 || 0 || 4
 | 0 || 0 || 0 || &amp;minus;4 || 0 || 0 
 |-
 ! 9 
 | 0 || 0 || &amp;minus;3 || 0 || 0 || &amp;minus;3 || 0 || 0 || style=&quot;background:#ffdead;&quot; | 6
 | 0 || 0 || &amp;minus;3 || 0 || 0 || &amp;minus;3 || 0 || 0 || 6
 | 0 || 0 || &amp;minus;3 || 0 || 0 || &amp;minus;3 || 0 || 0 || 6
 | 0 || 0 || &amp;minus;3  
 |-
 ! 10
 | 1 || &amp;minus;1 ||  1 || &amp;minus;1 || &amp;minus;4 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || style=&quot;background:#ffdead;&quot; | 4 
 | 1 || &amp;minus;1 ||  1 || &amp;minus;1 || &amp;minus;4 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || 4
 | 1 || &amp;minus;1 ||  1 || &amp;minus;1 || &amp;minus;4 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || 4
 |-
 ! 11
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 10
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || 10
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 
 |-
 ! 12
 | 0 || 2 || 0 || &amp;minus;2 || 0 || &amp;minus;4 || 0 || &amp;minus;2 || 0 || 2 || 0 || style=&quot;background:#ffdead;&quot; | 4
 | 0 || 2 || 0 || &amp;minus;2 || 0 || &amp;minus;4 || 0 || &amp;minus;2 || 0 || 2 || 0 || 4
 | 0 || 2 || 0 || &amp;minus;2 || 0 || &amp;minus;4 
 |-
 ! 13
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 12
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || 12
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 
 |-
 ! 14
 | 1 || &amp;minus;1 ||  1 || &amp;minus;1  ||  1 || &amp;minus;1 || &amp;minus;6 || &amp;minus;1 ||  1 || &amp;minus;1  ||  1 || &amp;minus;1 || 1 || style=&quot;background:#ffdead;&quot; | 6
 | 1 || &amp;minus;1 ||  1 || &amp;minus;1  ||  1 || &amp;minus;1 || &amp;minus;6 || &amp;minus;1 ||  1 || &amp;minus;1  ||  1 || &amp;minus;1 || 1 || 6
 | 1 || &amp;minus;1 
 |-
 ! 15
 | 1 || 1 || &amp;minus;2 || 1 || &amp;minus;4 || &amp;minus;2 || 1 || 1 || &amp;minus;2 || &amp;minus;4 || 1 || &amp;minus;2 || 1 || 1 || style=&quot;background:#ffdead;&quot; | 8
 | 1 || 1 || &amp;minus;2 || 1 || &amp;minus;4 || &amp;minus;2 || 1 || 1 || &amp;minus;2 || &amp;minus;4 || 1 || &amp;minus;2 || 1 || 1 || 8 
 |-
 ! 16
 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || &amp;minus;8  || 0 || 0 || 0 || 0 || 0 || 0 || 0 || style=&quot;background:#ffdead;&quot; | 8
 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || &amp;minus;8  || 0 || 0 || 0 || 0 || 0 || 0 
 |-
 ! 17
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 16
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1
 |-
 ! 18
 | 0 || 0 || 3 ||  0 || 0 || &amp;minus;3 ||  0 || 0 || &amp;minus;6 ||  0 || 0 || &amp;minus;3 ||  0 || 0 || 3 ||  0 || 0 || style=&quot;background:#ffdead;&quot; | 6
 | 0 || 0 || 3 ||  0 || 0 || &amp;minus;3 ||  0 || 0 || &amp;minus;6 ||  0 || 0 || &amp;minus;3  
 |-
 ! 19
 | &amp;minus;1 ||  &amp;minus;1 ||  &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 18
 | &amp;minus;1 ||  &amp;minus;1 ||  &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 
 |-
 ! 20
 | 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || &amp;minus;8 
 | 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || style=&quot;background:#ffdead;&quot; | 8 
 | 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || &amp;minus;8 
 |-
 ! 21
 | 1 || 1 || &amp;minus;2 || 1 || 1 || &amp;minus;2 || &amp;minus;6 || 1 || &amp;minus;2 || 1 || 1 || &amp;minus;2 || 1 || &amp;minus;6 || &amp;minus;2 || 1 || 1 || &amp;minus;2 || 1 || 1 || style=&quot;background:#ffdead;&quot; | 12
 | 1 || 1 || &amp;minus;2 || 1 || 1 || &amp;minus;2 || &amp;minus;6 || 1 || &amp;minus;2 
 |-
 ! 22
 | 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || &amp;minus;10
 | &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || style=&quot;background:#ffdead;&quot; | 10
 | 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1  
 |-
 ! 23
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 22
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 
 |-
 ! 24
 | 0 || 0 || 0 || 4 
 | 0 || 0 || 0 || &amp;minus;4
 | 0 || 0 || 0 || &amp;minus;8
 | 0 || 0 || 0 || &amp;minus;4
 | 0 || 0 || 0 || 4
 | 0 || 0 || 0 || style=&quot;background:#ffdead;&quot; | 8
 | 0 || 0 || 0 || 4 
 | 0 || 0 
 |-
 ! 25
 | 0 || 0 || 0 || 0 || &amp;minus;5
 | 0 || 0 || 0 || 0 || &amp;minus;5
 | 0 || 0 || 0 || 0 || &amp;minus;5
 | 0 || 0 || 0 || 0 || &amp;minus;5
 | 0 || 0 || 0 || 0 || style=&quot;background:#ffdead;&quot; | 20
 | 0 || 0 || 0 || 0 || &amp;minus;5
 |-
 ! 26
 | 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || &amp;minus;12
 | &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || 1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 12
 | 1 || &amp;minus;1 || 1 || &amp;minus;1 
 |-
 ! 27
 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || &amp;minus;9
 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || &amp;minus;9
 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || style=&quot;background:#ffdead;&quot; | 18
 | 0 || 0 || 0 
 |-
 ! 28
 | 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || &amp;minus;12
 | 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || &amp;minus;2 || 0 || 2 || 0 || style=&quot;background:#ffdead;&quot; | 12
 | 0 || 2
 |-
 ! 29
 | &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 28
 |  &amp;minus;1 
 |-
 !30
 | &amp;minus;1 ||  1 || 2 || 1 || 4 || &amp;minus;2 || &amp;minus;1 || 1 || 2 || &amp;minus;4 || &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || 1 || &amp;minus;8 
 | 1 ||  &amp;minus;1 || &amp;minus;2 || &amp;minus;1 || &amp;minus;4 || 2 || 1 || &amp;minus;1 || &amp;minus;2 || 4 || 1 || 2 || 1 || &amp;minus;1 || style=&quot;background:#ffdead;&quot; | 8 
 |}
 
 ==அடிக்குறிப்புகள்==
 
 {{reflist}}
 
 ==உசாத்துணையும் நூற்பட்டியலும்==
 
 *{{citation
   | last1 = Hardy  | first1 = G. H.
   | title = Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work
   | publisher = AMS / Chelsea
   | location = Providence RI
   | date = 1999
   | isbn = 978-0821820230}}
 
 *{{citation
   | last1 = Hardy  | first1 = G. H.
   | last2 = Wright | first2 = E. M. 
   | title = An Introduction to the Theory of Numbers (Fifth edition)
   | publisher = [[ஒக்ஸ்போர்ட் பல்கலைக்கழகப் பதிப்பகம்]]
   | location = Oxford
   | date = 1980
   | isbn = 978-0198531715}}
 
 *{{citation
   | last1 = Knopfmacher | first1 = John
   | title = Abstract Analytic Number Theory
   | publisher = Dover
   | location = New York
   | date = 1990
   | isbn = 0-486-66344-2}}
 
 *{{citation
   | title=Additive Number Theory: the Classical Bases  
   | volume=164 
   | series=Graduate Texts in Mathematics 
   | last=Nathanson 
   | first=Melvyn B. 
   | publisher=Springer-Verlag 
   | year=1996 | isbn=0-387-94656-X }}  Section A.7.
 
 *{{citation
   | last1 = Ramanujan  | first1 = Srinivasa
   | title = On Certain Trigonometric Sums and their Applications in the Theory of Numbers 
   | journal = Transactions of the Cambridge Philosophical Society
   | volume = 22
   | issue = 15
   | year = 1918
   | pages = 259&amp;ndash;276}} (pp.&amp;nbsp;179&amp;ndash;199 of his ''Collected Papers'')
 
 *{{citation
   | last1 = Ramanujan  | first1 = Srinivasa
   | title = On Certain Arithmetical Functions
   | journal = Transactions of the Cambridge Philosophical Society
   | volume = 22
   | issue = 9
   | year = 1916
   | pages = 159&amp;ndash;184}} (pp.&amp;nbsp;136&amp;ndash;163 of his ''Collected Papers'')
 
 *{{citation
   | last1 = Ramanujan  | first1 = Srinivasa
   | title = Collected Papers
   | publisher = AMS / Chelsea
   | location = Providence RI
   | date = 2000
   | isbn = 978-0821820766}}