Prime Numbers

1.5 Exercises 67
computationally efficient formula is
Z(t) =2
m
n −1/2 cos(t ln n − ϑ(t))
n=1
+(−1) m+1 τ −1/4
M
j=0
(−1) j τ −j/2 Φj(z)+RM (t).
Here, M is a cutoff integer of choice, the Φj are entire functions defined for
j ≥ 0 in terms of a function Φ0 and its derivatives, and RM (t) is the error. A
practical instance is the choice M = 2, for which we need
Φ0(z) =
cos( 1
2 πz2 + 3
8π) ,
cos(πz)
Φ1(z) = 1
Φ(3)
12π2 0 (z),
Φ2(z) = 1
1
Φ(2)
16π2 0 (z)+ Φ(6)
288π4 0 (z).
In spite of the complexity here, it is to be stressed that the formula is
immediately applicable in actual computation. In fact, the error R2 can be
rigorously bounded:
|R2(t)| < 0.011t −7/4
for all t>200.
Higher-order (M >2) bounds, primarily found in [Gabcke 1979], are known,
but just R2 has served computationalists well for two decades.
Implement the Riemann–Siegel formula for M = 2, and test against some
known values such as
ζ(1/2 + 300i) ≈ 0.4774556718784825545360619
+0.6079021332795530726590749 i,
Z(1/2 + 300i) ≈ 0.7729870129923042272624525,
which are accurate to the implied precision. Using your implementation, locate
the nearest zero to the point 1/2+300i, which zero should have t ≈ 299.84035.
You should also be able to find, still at the M = 2 approximation level and
with very little machine time, the value
ζ(1/2+10 6 i) ≈ 0.0760890697382 + 2.805102101019 i,
again correct to the implied precision.
When one is armed with a working Riemann–Siegel implementation, a
beautiful world of computation in support of analytic number theory opens.
For details on how actually to apply ζ evaluations away from the real axis,
see [Brent 1979], [van de Lune et al. 1986], [Odlyzko 1994], [Borwein et al.
2000]. We should point out that in spite of the power and importance of