கேள்வி: இராமானுசன் கூட்டு என்பது என்ன?
பதில்: :''இக்கட்டுரை இராமானுசன் கூட்டுகை (Ramanujan summation) என்பது பற்றியது அன்று'' கணிதத்தின் ஒரு பிரிவான எண்கோட்பாட்டியலில், இராமானுசன் கூட்டு (Ramanujan's sum), என்பதைப் பொதுவாக ''c'q''(''n''), எனக்குறிப்பது வழக்கம். இது நேர்ம எண் மாறிகள் ''q'', ''n'' ஆகியவற்றால் ஆன சார்பியம் (சார்பு). இதனைக் கீழ்க்காணும் சூத்திரத்தால் குறிக்கலாம் :c_q(n)= \sum_{a=1\atop (a,q)=1}^q e^{2 \pi i \tfrac{a}{q} n} , மேலுள்ளதில் (''a'', ''q'') = 1 என்னும் குறியீடு என்ன குறிக்கின்றதென்றால், ''a'' என்பது ''q'' என்னும் எண்ணோடு ஒப்பீட்டு பகா எண்ணின் (co-prime) மதிப்புகளை மட்டுமே கொள்ளும் என்று பொருள். இராமானுசன் இந்த கூட்டு வாய்பாட்டை 1918 ஆய்வுத்தாளில் அளித்தார்Ramanujan, ''On Certain Trigonometric Sums ...'' 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.(''Papers'', p. 179). In a footnote cites pp. 360&ndash;370 of the Dirichlet-Dedekind ''Vorlesungen über Zahlentheorie'', 4th ed. இக் கூட்டு வாய்பாட்டினை வினோகிராடோவ் தேற்றத்தை(Vinogradov's theorem)நிறுவுவதில் பயன்படுத்தியுள்ளார்கள். இத்தேற்றம் மிகப்பெரிய ஒற்றைப்படை எண்கள் ஒவ்வொன்றும் மூன்று பகா எண்களின் கூட்டுத்தொகை என கூறுகின்றதுNathanson, ch. 8
கேள்வி: இராமானுசன் கூட்டு - இராமானுசன் கூட்டு அட்டவணை பற்றி தகவல் சொல்.
பதில்: {| class="wikitable" style="text-align:center" cellpadding="2" |+ Ramanujan Sum ''c's''(''n'') |- | colspan="2" rowspan="2" | &nbsp; ! colspan="30" | n |- ! width="25" | 1 ! width="25" | 2 ! width="25" | 3 ! width="25" | 4 ! width="25" | 5 ! width="25" | 6 ! width="25" | 7 ! width="25" | 8 ! width="25" | 9 ! width="25" | 10 ! width="25" | 11 ! width="25" | 12 ! width="25" | 13 ! width="25" | 14 ! width="25" | 15 ! width="25" | 16 ! width="25" | 17 ! width="25" | 18 ! width="25" | 19 ! width="25" | 20 ! width="25" | 21 ! width="25" | 22 ! width="25" | 23 ! width="25" | 24 ! width="25" | 25 ! width="25" | 26 ! width="25" | 27 ! width="25" | 28 ! width="25" | 29 ! width="25" | 30 |- ! rowspan="30" width="25" | s ! width="25" | 1 | style="background:#ffdead;" | 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 | &minus;1 | style="background:#ffdead;" | 1 | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 |- ! 3 | &minus;1 || &minus;1 | style="background:#ffdead;" | 2 | &minus;1 || &minus;1 || 2 | &minus;1 || &minus;1 || 2 || &minus;1 || &minus;1 || 2 | &minus;1 || &minus;1 || 2 || &minus;1 || &minus;1 || 2 | &minus;1 || &minus;1 || 2 || &minus;1 || &minus;1 || 2 | &minus;1 || &minus;1 || 2 || &minus;1 || &minus;1 || 2 |- ! 4 | 0 || &minus;2 || 0 | style="background:#ffdead;" | 2 | 0 || &minus;2 || 0 || 2 || 0 || &minus;2 | 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 | 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 |- ! 5 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || style="background:#ffdead;" | 4 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 4 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 4 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 4 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 4 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 4 |- ! 6 | 1 || &minus;1 || &minus;2 || &minus;1 || 1 || style="background:#ffdead;" | 2 | 1 || &minus;1 || &minus;2 || &minus;1 || 1 || 2 | 1 || &minus;1 || &minus;2 || &minus;1 || 1 || 2 | 1 || &minus;1 || &minus;2 || &minus;1 || 1 || 2 | 1 || &minus;1 || &minus;2 || &minus;1 || 1 || 2 |- ! 7 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || style="background:#ffdead;" | 6 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 6 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 6 | &minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 ||&minus;1 || 6 | &minus;1 ||&minus;1 |- ! 8 | 0 || 0 || 0 || &minus;4 || 0 || 0 || 0 || style="background:#ffdead;" | 4 | 0 || 0 || 0 || &minus;4 || 0 || 0 || 0 || 4 | 0 || 0 || 0 || &minus;4 || 0 || 0 || 0 || 4 | 0 || 0 || 0 || &minus;4 || 0 || 0 |- ! 9 | 0 || 0 || &minus;3 || 0 || 0 || &minus;3 || 0 || 0 || style="background:#ffdead;" | 6 | 0 || 0 || &minus;3 || 0 || 0 || &minus;3 || 0 || 0 || 6 | 0 || 0 || &minus;3 || 0 || 0 || &minus;3 || 0 || 0 || 6 | 0 || 0 || &minus;3 |- ! 10 | 1 || &minus;1 || 1 || &minus;1 || &minus;4 || &minus;1 || 1 || &minus;1 || 1 || style="background:#ffdead;" | 4 | 1 || &minus;1 || 1 || &minus;1 || &minus;4 || &minus;1 || 1 || &minus;1 || 1 || 4 | 1 || &minus;1 || 1 || &minus;1 || &minus;4 || &minus;1 || 1 || &minus;1 || 1 || 4 |- ! 11 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 10 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || 10 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 |- ! 12 | 0 || 2 || 0 || &minus;2 || 0 || &minus;4 || 0 || &minus;2 || 0 || 2 || 0 || style="background:#ffdead;" | 4 | 0 || 2 || 0 || &minus;2 || 0 || &minus;4 || 0 || &minus;2 || 0 || 2 || 0 || 4 | 0 || 2 || 0 || &minus;2 || 0 || &minus;4 |- ! 13 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 12 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || 12 | &minus;1 || &minus;1 || &minus;1 || &minus;1 |- ! 14 | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &minus;6 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || style="background:#ffdead;" | 6 | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &minus;6 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || 6 | 1 || &minus;1 |- ! 15 | 1 || 1 || &minus;2 || 1 || &minus;4 || &minus;2 || 1 || 1 || &minus;2 || &minus;4 || 1 || &minus;2 || 1 || 1 || style="background:#ffdead;" | 8 | 1 || 1 || &minus;2 || 1 || &minus;4 || &minus;2 || 1 || 1 || &minus;2 || &minus;4 || 1 || &minus;2 || 1 || 1 || 8 |- ! 16 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || &minus;8 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || style="background:#ffdead;" | 8 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || &minus;8 || 0 || 0 || 0 || 0 || 0 || 0 |- ! 17 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 16 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 |- ! 18 | 0 || 0 || 3 || 0 || 0 || &minus;3 || 0 || 0 || &minus;6 || 0 || 0 || &minus;3 || 0 || 0 || 3 || 0 || 0 || style="background:#ffdead;" | 6 | 0 || 0 || 3 || 0 || 0 || &minus;3 || 0 || 0 || &minus;6 || 0 || 0 || &minus;3 |- ! 19 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 18 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 |- ! 20 | 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || &minus;8 | 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || style="background:#ffdead;" | 8 | 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || &minus;8 |- ! 21 | 1 || 1 || &minus;2 || 1 || 1 || &minus;2 || &minus;6 || 1 || &minus;2 || 1 || 1 || &minus;2 || 1 || &minus;6 || &minus;2 || 1 || 1 || &minus;2 || 1 || 1 || style="background:#ffdead;" | 12 | 1 || 1 || &minus;2 || 1 || 1 || &minus;2 || &minus;6 || 1 || &minus;2 |- ! 22 | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &minus;10 | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || style="background:#ffdead;" | 10 | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 |- ! 23 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 22 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 |- ! 24 | 0 || 0 || 0 || 4 | 0 || 0 || 0 || &minus;4 | 0 || 0 || 0 || &minus;8 | 0 || 0 || 0 || &minus;4 | 0 || 0 || 0 || 4 | 0 || 0 || 0 || style="background:#ffdead;" | 8 | 0 || 0 || 0 || 4 | 0 || 0 |- ! 25 | 0 || 0 || 0 || 0 || &minus;5 | 0 || 0 || 0 || 0 || &minus;5 | 0 || 0 || 0 || 0 || &minus;5 | 0 || 0 || 0 || 0 || &minus;5 | 0 || 0 || 0 || 0 || style="background:#ffdead;" | 20 | 0 || 0 || 0 || 0 || &minus;5 |- ! 26 | 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &minus;12 | &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || 1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 12 | 1 || &minus;1 || 1 || &minus;1 |- ! 27 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || &minus;9 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || &minus;9 | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || style="background:#ffdead;" | 18 | 0 || 0 || 0 |- ! 28 | 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || &minus;12 | 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || &minus;2 || 0 || 2 || 0 || style="background:#ffdead;" | 12 | 0 || 2 |- ! 29 | &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || &minus;1 || style="background:#ffdead;" | 28 | &minus;1 |- !30 | &minus;1 || 1 || 2 || 1 || 4 || &minus;2 || &minus;1 || 1 || 2 || &minus;4 || &minus;1 || &minus;2 || &minus;1 || 1 || &minus;8 | 1 || &minus;1 || &minus;2 || &minus;1 || &minus;4 || 2 || 1 || &minus;1 || &minus;2 || 4 || 1 || 2 || 1 || &minus;1 || style="background:#ffdead;" | 8 |}
கேள்வி: இராமானுசன் கூட்டு - உசாத்துணையும் நூற்பட்டியலும் பற்றி தகவல் சொல்.
பதில்: * * * * Section A.7. * (pp.&nbsp;179&ndash;199 of his ''Collected Papers'') * (pp.&nbsp;136&ndash;163 of his ''Collected Papers'') * பகுப்பு:எண் கோட்பாடு