Prime Numbers

1.5 Exercises 61
Then using Exercise 1.37 give a proof that the Riemann hypothesis is
equivalent to the elementary assertion
|L(n) − n| < √ n ln 2 n for every integer n ≥ 3, (1.49)
where L(n) is the natural logarithm of the least common multiple of 1, 2,...,n.
If (1.48) is to be the “calculus-course” version of the Riemann hypothesis,
perhaps (1.49) might be referred to as the “precalculus-course” version, in
that all that is used in the formulation here is the concept of least common
multiple and the natural logarithm.
1.39. Using the conjectured form of the PNT in (1.25), prove that
there is a prime between every pair of sufficiently large cubes. Use (1.48)
and any relevant computation to establish that (again, on the Riemann
hypothesis) there is a prime between every two positive cubes. It was shown
unconditionally by Ingham in 1937 that there is a prime between every pair
of sufficiently large cubes, and it was shown, again unconditionally, by Cheng
in 1999, that this is true for cubes greater than e e15
.
1.40. Show that
p≤n−2 1/ ln(n − p) ∼ n/ ln2 n, where the sum is over
primes.
1.41. Using the known theorem that there is a positive number c such that
the number of even numbers up to x that cannot be represented as a sum of
two primes is O(x 1−c ), show that there are infinitely many triples of primes in
arithmetic progression. (For a different approach to the problem, see Exercise
1.42.)
1.42. It is known via the theory of exponential sums that
n≤x
(R2(2n) −R2(2n)) 2 3 x
= O
ln 5
, (1.50)
x
where R2(2n) is, as in the text, the number of representations p+q =2n with
p, q prime, and where R2(2n) is given by (1.10); see [Prachar 1978]. Further,
we know from the Brun sieve method that
n ln ln n
R2(2n) =O
ln 2
.
n
Show, too, that R2(2n) enjoys the same big-O relation. Use these estimates to
prove that the set of numbers 2p with p prime and with 2p not representable
as a sum of two distinct primes has relative asymptotic density zero in the set
of primes; that is, the number of these exceptional primes p ≤ x is o(π(x)).
In addition, let
A3(x) =# (p, q, r) ∈P 3 : 0