Prime Numbers

158 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
block, and we also compute φ(r2k ,pb) foreachb, so as to use these for the
next block. The computed values of φ(x/(mpb+1),pb) are not stored, but are
multiplied by µ(m) and added into a running sum that represents the second
term on the right of (3.23). The time and space required to do these tasks
for all pb ≤ x 1/3 in the r-th block is O(x 1/3+ɛ ). The values of φ r2 k ,pb
are written over the prior values φ((r − 1)2 k ,pb), so the total space used is
O x 1/3+ɛ . The total number of blocks does not exceed x 1/3 , so the total time
used in this computation is O x 2/3+ɛ , as advertised.
There are various ideas for speeding up this algorithm in practice, see
[Lagarias et al. 1985] and [Deléglise and Rivat 1996].
3.7.2 Analytic method
Here we describe an analytic method, highly efficient in principle, for counting
primes. The idea is that in [Lagarias and Odlyzko 1987], with recent extensions
that we shall investigate. The idea is to exploit the fact that the Riemann zeta
function embodies in some sense the properties of primes. A certain formal
manipulation of the Euler product relation (1.18) goes like so. Start by taking
the logarithm
ln ζ(s) =ln
p∈P
and then introduce a logarithmic series
(1 − p −s ) −1 = −
ln ζ(s) =
∞
p∈P m=1
p∈P
ln(1 − p −s ),
1
, (3.24)
mpsm where all manipulations are valid (and the double sum can be interchanged
if need be) for Re(s) > 1, with the caveat that ln ζ is to be interpreted as
a continuously changing argument. (By modern convention, one starts with
the positive real ln ζ(2) and tracks the logarithm as the angle argument of ζ,
along a contour that moves vertically to 2 + i Im(s) thenovertos.)
In order to use relation (3.24) to count primes, we define a function
reminiscent of—but not quite the same as—the prime-counting function π(x).
In particular, we consider a sum over prime powers not exceeding x, namely
π ∗ (x) =
p∈P, m>0
θ(x − pm )
, (3.25)
m
where θ(z) is the Heaviside function, equal to 1, 1/2, 0, respectively, as its
argument z is positive, zero, negative. The introduction of θ means that the
sum involves only prime powers p m not exceeding x, but that whenever the
real x actually equals a power p m , the summand is 1/(2m). The next step is
to invoke the Perron formula, which says that for nonnegative real x, positive
integer n, and a choice of contour C = {s :Re(s) =σ}, with fixed σ>0and