Prime Numbers

3.7 Counting primes 161
Indeed, the last summation is rather easy, since it has just O ( √ x)terms.The
next-to-last summation, which just records the difference between π(x) and
π ∗ (x), also has just O ( √ x)terms.
Let us posit a specific smooth decay, i.e., for u ∈ (x − y, x] we define
(x − u)2
c(u, x) =3
y2 (x − u)3
− 2
y3 .
Observe that c(x − y, x) =1andc(x, x) = 0, as required for continuous c
functions in the stated class. Mellin transformation of c gives
y3 F (s, x) = (3.30)
6
−2xs+3 +(s +3)xs+2y +(x− y) s (2x3 +(s− 3)x2y − 2sxy2 +(s +1)y3 )
.
s(s + 1)(s + 2)(s +3)
This expression, though rather unwieldy, allows us to count primes more
efficiently. For one thing, the denominator of the second fraction is O(t 4 ),
which is encouraging. As an example, performing numerical integration as in
relation (3.29) with the choices x = 100,y = 10, we find for the same trial set
of integration limits T ∈{10, 30, 50, 70, 90} the results
π(100) ≈ 25.3, 26.1, 25.27, 24.9398, 24.9942,
which are quite satisfactory, since π(100) = 25. (Note, however, that there is
still some chaotic behavior until T be sufficiently large.) It should be pointed
out that Lagarias and Odlyzko suggest a much more general, parameterized
form for the Mellin pair c, F , and indicate how to optimize the parameters.
Their complexity result is that one can either compute π(x) with bit operation
count O x 1/2+ɛ and storage space of O x 1/4+ɛ bits, or on the notion of
limited memory one may replace the powers with 3/5+ɛ, ɛ, respectively.
As of this writing, there has been no practical result of the analytic method
on a par with the greatest successes of the aforementioned combinatorial
methods. However, this impasse apparently comes down to just a matter of
calendar time. In fact, [Galway 1998] has reported that values of π(10 n )for
n = 13, and perhaps 14, are attainable for a certain turn-off function c and
(only) standard, double-precision floating-point arithmetic for the numerical
integration. Perhaps 100-bit or higher precision will be necessary to press the
analytic method on toward modern limits, say x ≈ 10 21 or more; the required
precision depends on detailed error estimates for the contour integral. The
Galway functions are a clever choice of Mellin pair, and work out to be more
efficient than the turn-off functions that lead to F of the type (3.30). Take
c(u, x) = 1
2 erfc
ln u
x
2a(x)
,
where erfc is the standard error function:
erfc(z) = 2
∞
√ e
π
−t2
dt
z