1.3 The Farey sequence 105
The Farey polygonsP 10 andP 20
Letφ(n)be the number of integersmsatisfying 1 ≤ m ≤ nand
gcd(m, n)=1. The Farey sequenceFnhas1+
∑n
k=1φ(k)terms. The
polygonPncontains2+
∑n
k=1φ(k)boundary points and no interior
points. By Pick’s formula, its area isAn=^12
∑n
k=1φ(k). By a calcula-
tion of D. N. Lehmer [Lehmer], for large values ofn, this area is about
3
2 π^2 of the (smallest) square containing it:
lim
n→∞An
n^2=
3
2 π^2