I read on the Wikipedia page for the ramanujan sum a formula for the sum of 2 squares in terms of the ramanujan sums. But I did not get it.


How do ramanujan sums and the sum of 2 squares relate ? How to prove that identity ?

  • $\begingroup$ See mathworld.wolfram.com/SumofSquaresFunction.html, formulas $(20)-(30)$ $\endgroup$ – Dietrich Burde Jul 22 '14 at 11:41
  • $\begingroup$ @DietrichBurde that page does not even mention ramanujan sums !? $\endgroup$ – mick Jul 22 '14 at 19:42
  • $\begingroup$ As the Ramanujan expansion section of the Wikipedia article suggests, there may be no special relationship between Ramanujan sums and expressing integers as sums-of-squares. Rather, arithmetic functions in general have a kind of Fourier expansion in terms of Ramanujan sums, and the sums-of-squares counting function is a particular kind of arithmetic function. For general background I'd suggest W. Schwarz, J. Spilker (1994) and anything they cite in the chapter on Ramanujan expansions. If there is any special connection, it'd likely be through theta functions or the like. $\endgroup$ – blue Jul 23 '14 at 1:41

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