Prime Numbers

1.5 Exercises 53
1.17. It can happen that a polynomial, while not always producing primes,
is very likely to do so over certain domains. Show by computation that a
polynomial found in [Dress and Olivier 1999],
f(x) =x 2 + x − 1354363,
has the astounding property that for a random integer x ∈ [1, 10 4 ], the number
|f(x)| is prime with probability exceeding 1/2. An amusing implication is this:
If you can remember the seven-digit “phone number” 1354363, then you have
a mental mnemonic for generating thousands of primes.
1.18. Consider the sequence of primes 2, 3, 5, 11, 23, 47. Each but the first
is one away from the double of the prior prime. Show that there cannot be
an infinite sequence of primes with this property, regardless of the starting
prime.
1.19. As mentioned in the text, the relation
1
ζ(s) =
∞
n=1
µ(n)
n s
is valid (the sum converges absolutely) for Re(s) > 1. Prove this. But the
limit as s → 1, for which we know the remarkable PNT equivalence
∞
n=1
µ(n)
n =0,
is not so easy. Two good exercises are these: First, via numerical experiments,
furnish an estimate for the order of magnitude of
n≤x
µ(n)
n
as a function of x; and second, provide an at least heuristic argument as to
why the sum should vanish as x →∞. For the first option, it is an interesting
computational challenge to work out an efficient implementation of the µ
function itself. As for the second option, you might consider the first few
terms in the form
1 −
p≤x
1 1
+
p pq
pq≤x
−···
to see why the sum tends to zero for large x. It is of interest that even without
recourse to the PNT, one can prove, as J. Gram did in 1884 [Ribenboim 1996],
that the sum is bounded as x →∞.
1.20. Show that for all x>1, we have
p≤x
1
> ln ln x − 1,
p