Prime Numbers

162 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
and a is chosen later for efficiency. This c function turns off smoothly at u ∼ x,
but at a rate tunable by choice of a. The Mellin companion works out nicely
to be
F (s) = xs
s es2 a(x) 2
. (3.31)
For s = σ+it the wonderful (for computational purposes) decay in F is e −t2 a 2
.
Now numerical experiments are even more satisfactory. Sure enough, we can
use relation (3.29) to yield, for x = 1000, decay function a(x) =(2x) −1/2 ,
σ =3/2, and integration limits T ∈{20, 40, 60, 80, 100, 120}, the successive
values
π(1000) ≈ 170.6, 169.5, 170.1, 167.75, 167.97, 167.998,
in excellent agreement with the exact value π(1000) = 168; and furthermore,
during such a run the chaotic manner of convergence is, qualitatively speaking,
not so manifest.
Incidentally, though we have used properties of ζ(s) totherightofthe
critical strip, there are ways to count primes using properties within the strip;
see Exercise 3.50.
3.8 Exercises
3.1. In the spirit of the opening observations to the present chapter, denote
by SB(n) the sum of the base-B digits of n. Interesting phenomena accrue for
specific B, suchasB = 7. Find the smallest prime p such that S7(p) is itself
composite. (The magnitude of this prime might surprise you!) Then, find all
possible composite values of S7(p) for the primes p