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**Example text**

Prove that µ(n) is multiplicative. Deduce the formula 1 if n = 1, 0 otherwise. µ(d) = d|n Hint: By the multiplicativity it suﬃces to prove the latter identity only when n is the power of a prime. * This exercise is a continuation of the preceding one. i) For Euler’s totient, show that ϕ(n) = n d|n µ(d) =n d 1− p|n 1 p . Deduce that ϕ(n) = n≤N bµ(d) = bd≤N N2 2 ∞ d=1 µ(d) + O(N log N ). 19) ζ(2) := π2 1 ; = n2 6 n=1 the convergence of this inﬁnite series is clear by Riemann’s convergence criterion.

Q. By the pigeonhole principle there has to be at least one interval which contains at least two numbers {kα} ≥ { α}, say, with 0 ≤ k, ≤ Q and k = . It follows that {kα} − { α} = kα − [kα] − α + [ α] = {(k − )α} + [(k − )α] + [ α] − [kα] . 2) add Since {kα} − { α} lies in the interval [0, Q up to zero. Setting q = k − we obtain {qα} = {kα} − { α} < 1 . 1) (since q < Q). Now suppose that α is irrational and that there exist only ﬁnitely many solutions pq11 , . . 1). Since α ∈ Q, we can ﬁnd a Q such that α− pj 1 > qj Q for j = 1, .

It is clear that 1 x a − = . 12) Now suppose that c d < xy ; then c dx − cy 1 x − = ≥ . y d dy dy Further, a bc − ad 1 c − = ≥ . d b bd bd All these estimates imply x c c a 1 1 y+b n x a − = − + − ≥ + = > . 12) it follows that n < d, contradicting c x d = y which proves the theorem. • c d ∈ Fn . 8. Mediants and Ford circles The proof of the previous theorem gives a rule for the computation of the successor of a Farey fraction ab in Fn . This successor is also related to the former right neighbor of ab .