On amicable numbers by Euler L.
C be a function satisfying the conditions (V1), (V2), and (V3). Then L xEr f(x) = vol Proof. We assume first that r zn. (y). , F(u Hence it can be developed into a Fourier series L + x) = F(x) for all x E zn. e27riu'Ya(y), yElP where a(y):= J F(t)e-27riyot dt. We shall show that [O,l]n a(y) = i(y). Then condition (V3) implies that the Fourier series of F converges absolutely and uniformly, hence it converges to a continuous function, hence to F.
K! k· Proof We consider the function T definition of Bk yields 7rT cotg7rT . f----+ e 7ri 7" 7rZT. e7rt7" . 7rZT 1 7rT cotg 7rT. Then putting x + e- 7ri7" _ . e- 7rt7" 27riT + e 27ri 7" _ . - e 27rt 7" - 1 _ ~ B (27riT)k 1 - 1+ ~ k k! k=2 ~ B (27ri)k k +~ k~T. 4. 1 With the Bernoulli numbers Bk one has Proof. 4, the coefficient in front of the infinite sum is equal to (k - I)! 2((k) (27ri)k 2k! (k - I)! (27ri)k Bk 2k - Bk . This proves the corollary. So, for example, r=l 00 1 - 504 I>·5(r)qr.