Prime Numbers

68 Chapter 1 PRIMES!
the Riemann–Siegel formula, there are yet alternative means for efficient
evaluation when imaginary parts are large. In fact it is possible to avoid
the inherently asymptotic character of the Riemann–Siegel series, in favor of
manifestly convergent expansions based on incomplete gamma function values,
or on saddle points of certain integrals. Alternative schemes are discussed in
[Galway 2000], [Borwein et al. 2000], and [Crandall 1999c].
1.62. For the Riemann–Siegel formula of Exercise 1.61, and for similar
prescriptions when s = σ + it is not on the half-line, it is evident that sums
of the form
Sm(s) =
m
n=1
1
,
ns where m is an appropriate cutoff (typically, m ∼ √ t), could be used in actual
computations. Investigate the notion of calculating Sm(s) overanarithmetic
progression of s values, using the nonuniform FFT algorithm we present as
Algorithm 9.5.8. That is, for values
for say k =0,...,K− 1, we have
Sm(σ + ikτ) =
s = σ + ikτ,
m
n=1
1
n σ e−ikτ ln n ,
and sure enough, this suggests a strategy of (m/K) nonuniform FFTs each of
length K. Happily, the sum Sm can thus be calculated, for all k ∈ [0,K− 1],
in a total of
O(m ln K)
operations, where desired accuracy enters (only logarithmically) into the
implied big-O constant. This is a remarkable gain over the naive approach
of doing a length-m sum K times, which would require O(mK).
Such speedups can be used not only for RH verification, but analytic
methods for prime-counting. Incidentally, this nonuniform FFT approach
is essentially equivalent in complexity to the parallel method in [Odlyzko
and Schönhage 1988]; however, for computationalists familiar with FFT, or
possessed of efficient FFT software (which the nonuniform FFT could call
internally), the method of the present exercise should be attractive.
1.63. Show that ψ(x), defined in (1.22), is the logarithm of the least
common multiple of all the positive integers not exceeding x. Show
that the prime number theorem is equivalent to the assertion ψ(x) ∼
x. Incidentally, in [Deléglise and Rivat 1998], ψ 10 15 is found to be
999999997476930.507683 ..., an attractive numerical instance of the relation
ψ(x) ∼ x. We see, in fact, that the error |ψ(x) − x| is very roughly √ x for
x =10 15 , such being the sort of error one expects on the basis of the Riemann
hypothesis.