20 1. Elementary Prime Number Theory, I

e2 e1

µ ξ

f

e

√

|D|

2

Figure 3. (Based on [Zau83].)

Figure 2. Then for either ξ = 1 or ξ = −1, we have

|ξ − π1/ρ1| 1.

Then (P1) and (P2) hold if we choose this value of ξ and take γ = 1. Note

that π1 ±ρ1, since otherwise π1 and ρ1 would be unit multiples of each

other, which we have already argued is not the case.

So we may assume that π1/ρ1 lies within the shaded region. Let e1 be

the ray from the origin making an angle of

60◦

with the x-axis, and let e2

be the ray from the origin making an angle of

120◦

with that axis. Then the

ray e (say) from the origin through π1/ρ1 is contained within the

60◦

angle

determined by e1 and

e2.2

Let f be the horizontal line consisting of those

complex numbers with imaginary part |D|/2; thus f is the first horizontal

line above the x-axis containing points of the lattice Z + Zη. Let µ be the

2Here

the angle determined by e1 and e2 means the closed set of points between e1 and e2.