Prime Numbers

1.5 Exercises 59
As an extension, use exponential sums to study the count
πc(x) =#{n ∈ [1,x]:⌊n c ⌋∈P}.
Heuristically, one might expect the asymptotic behavior
πc(x) ∼ 1
c
x
ln x .
Show first, on the basis of the PNT, that for c ≤ 1 this asymptotic relation
indeed holds. Use exponential sum techniques to establish this asymptotic
behavior for some c>1; for example, there is the Piatetski-Shapiro theorem
[Graham and Kolesnik 1991] that the asymptotic relation holds for any c with
1 1 one takes the “principal value” for the
singularity of the integrand at t =1,namely,
1−ɛ x
1
1
li0(x) = lim
dt +
ɛ→0 ln t ln t dt
.
0
The function li0(x) isli(x)+c, wherec ≈ 1.0451637801. Before Skewes came
up with his bounds, J. Littlewood had shown that π(x) − li0(x) (as well as
π(x) − li (x)) not only changes sign, but does so infinitely often.
An amusing first foray into the “Skewes world” is to express the second
Skewes number above in decimal-exponential notation (in other words, replace
the e’s with 10’s appropriately, as has been done already for the first Skewes
number). Incidentally, a newer reference on the problem is [Kaczorowski
1984], while a modern estimate for the least x with π(x) > li0(x) is
x li0(x) for some
x ∈ (1.398201, 1.398244) · 10 316 .
One interesting speculative exercise is to estimate roughly how many more
years it will take researchers actually to find and prove an explicit case of
π(x) > li0(x). It is intriguing to guess how far calculations of π(x) itself can
be pushed in, say, 30 years. We discuss prime-counting algorithms in Section
3.7, although the state of the art is today π 10 21 or somewhat higher than
this (with new results emerging often).
Another speculative direction: Try to imagine numerical or even physical
scenarios in which such huge numbers naturally arise. One reference for
1+ɛ