Prime Numbers

58 Chapter 1 PRIMES!
Exercise 1.35. Now think of the ordered, natural logarithms of the Fermat
numbers as a pseudorandom sequence of real numbers. Prove this theorem: If
said sequence is equidistributed modulo 1, then the number ln 2 is normal to
base 2. Is the converse of this theorem true?
Note that it remains unknown to this day whether ln 2 is normal to any
integer base. Unfortunately, the same can be said for any of the fundamental
constants of history, such as π, e, and so on. That is, except for instances
of artificial digit construction as in Exercise 1.32, normality proofs remain
elusive. A standard reference for rigorous descriptions of normality and
equidistribution is [Kuipers and Niederreiter 1974]. A discussion of normality
properties for specific fundamental constants such as ln 2 is [Bailey and
Crandall 2001].
1.34. Using the PNT, or just Chebyshev’s Theorem 1.1.3, prove that the
set of rational numbers p/q with p, q prime is dense in the positive reals.
1.35. It is a theorem of Vinogradov that for any irrational number α,
the sequence (αpn), where the pn are the primes in natural order, is
equidistributed modulo 1. Equidistribution here means that if #(a, b, N)
denotes the number of times any interval [a, b) ⊂ [0, 1) is struck after N
primes are used, then #(a, b, N)/N ∼ (b − a) asN →∞. On the basis of this
Vinogradov theorem, prove the following: For irrational α>1, and the set
the prime count defined by
behaves as
S(α) ={⌊kα⌋ : k =1, 2, 3,...},
πα(x) =#{p ≤ x : p ∈P∩S(α)}
πα(x) ∼ 1 x
α ln x .
What is the behavior of πα for α rational?
As an extension to this exercise, the Vinogradov equidistribution theorem
itself can be established via the exponential sum ideas of Section 1.4.4. One
uses the celebrated Weyl theorem on spectral properties of equidistributed
sequences [Kuipers and Niederreiter 1974, Theorem 2.1] to bring the problem
down to showing that for irrational α and any integer h = 0,
EN(hα) =
e 2πihαp
p≤N
is o(N). This, in turn, can be done by finding suitable rational approximants
to α and providing bounds on the exponential sum, using essentially our book
formula (1.39) for well-approximable values of hα, while for other α using
(1.41). The treatment in [Ellison and Ellison 1985] is pleasantly accessible on
this matter.