20 1. Elementary Prime Number Theory, I
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
with the x-axis, and let e2
be the ray from the origin making an angle of
with that axis. Then the
ray e (say) from the origin through π1/ρ1 is contained within the
determined by e1 and
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
the angle determined by e1 and e2 means the closed set of points between e1 and e2.