Prime Numbers

52 Chapter 1 PRIMES!
This kind of combinatorial reasoning can be used, as Legendre once did, to
show that π(x) =o(x). To that end, show that
φ(x, y) =x
1 − 1
+ E,
p
p≤y
where the error term E is O(2 π(y) ). Now use this last relation and the fact that
the sum of the reciprocals of the primes diverges to argue that π(x)/x → 0as
x →∞. (Compare with Exercise 1.11.)
1.14. Starting with the fundamental Theorem 1.1.1, show that for any fixed
ɛ>0, the number d(n) of divisors of n (including always 1 and n) satisfies
d(n) =O(n ɛ ).
How does the implied O-constant depend on the choice of ɛ? You might get
started in this problem by first showing that for fixed ɛ, there are only finitely
many prime powers q with d(q) >q ɛ .
1.15. Consider the sum of the reciprocals of all Mersenne numbers Mn =
2 n − 1 (for positive integers n), namely,
E =
∞
1
Mn
n=1
Prove the following alternative form involving the divisor function d (defined
in Exercise 1.14):
∞ d(k)
E = .
2k k=1
Actually, one can give this sum a faster-than-linear convergence. To that end
show that we also have
∞ 1
E =
2m2 2m +1
2m − 1 .
m=1
Incidentally, the number E has been well studied in some respects. For
example, it is known [Erdős 1948], [Borwein 1991] that E is irrational, yet
it has never been given a closed form. Possible approaches to establishing
deeper properties of the number E are laid out in [Bailey and Crandall 2002].
If we restrict such a sum to be over Mersenne primes, then on the basis of
Table 1.2, and assuming that said table is exhaustive up through its final entry
(note that this is not currently known), to how many good decimal digits do
we know
1
?
Mq
Mq∈P
1.16. Euler’s polynomial x 2 +x+41 has prime values for each integer x with
−40 ≤ x ≤ 39. Show that if f(x) is a nonconstant polynomial with integer
coefficients, then there are infinitely many integers x with f(x) composite.
.