Prime Numbers

1.5 Exercises 65
1.56. Prove the following theorem relevant to Wilson primes: if g is a
primitive root of the prime p, then the Wilson quotient is given by
j g
p−1
wp ≡
j=1
p
g p−1−j (mod p).
Then, using this result, give an algorithm that determines whether p with
primitive root g = 2 is a Wilson prime, but using no multiplications; merely
addition, subtraction, and comparison.
1.57. There is a way to connect the notion of twin-prime pairs with the
Wilson–Lagrange theorem as follows. Let p be an integer greater than 1. Prove
the theorem of Clement that p, p + 2 is a twin-prime pair if and only if
4(p − 1)! ≡−4 − p (mod p(p + 2)).
1.58. How does one resolve the following “Mertens paradox”? Say x is a
large integer and consider the “probability” that x is prime. As we know,
primality
√
can be determined by testing x for prime divisors not exceeding
x. But from Theorem 1.4.2, it would seem that when all the primes less
than √ x are probabilistically sieved out, we end up with probability
p≤ √ x
1 − 1
p
∼ 2e−γ
ln x .
Arrive again at this same estimate by simply removing the floor functions in
(1.46). However, the PNT says that the correct asymptotic probability that
x is prime is 1/ ln x. Note that 2e −γ =1.1229189 ..., so what is a resolution?
It has been said that the sieve of Eratosthenes is “more efficient than
random,” and that is one way to envision the “paradox.” Actually, there has
been some interesting work on ways to think of a resolution; for example, in
[Furry 1942] there is an analysis of the action of the sieve of Eratosthenes on a
prescribed interval [x, x + d], with some surprises uncovered in regard to how
many composites are struck out of said interval; see [Bach and Shallit 1996,
p. 365] for a historical summary.
1.59. By assuming that relation (1.24) is valid whenever the integral
converges, prove that M(x) =O(x 1/2+ɛ ) implies the Riemann hypothesis.
1.60. There is a compact way to quantify the relation between the PNT and
the behavior of the Riemann zeta function. Using the relation
− ζ′ ∞
(s)
= s ψ(x)x
ζ(s) −s−1 dx,
show that the assumption
1
ψ(x) =x + O(x α )