Prime Numbers

60 Chapter 1 PRIMES!
this recreation is [Crandall 1997a]. In that reference, what might be called
preposterous physical scenarios—such as the annual probability of finding
oneself accidentally quantum-tunneled bodily (and alive, all parts intact!) to
planet Mars—are still not much smaller than A −A ,whereA is the Avogadro
number (a mole, or about 6·10 23 ). It is difficult to describe a statistical scenario
relevant to the primes that begs of yet higher exponentiation as manifest in
theSkewesnumber.
Incidentally, for various technical reasons, the logarithmic-integral function
li0, on many modern numerical/symbolic systems, is best calculated in
terms of Ei(ln x), where we refer to the standard exponential-integral function
Ei(z) =
z
t
−∞
−1 e t dt,
with principal value assumed for the singularity at t = 0. In addition, care
must be taken to observe that some authors use the notation li for what we
are calling li0, rather than the integral from 2 in our defining equation (1.3)
for li . Calling our book’s function li , and the latter li0, we can summarize
this computational advice as follows:
li (x) =li0(x) − li0(2) = Ei(ln x) − Ei(ln 2) ≈ Ei(ln x) − 1.0451637801.
1.37. In [Schoenfeld 1976] it is shown that on the Riemann hypothesis we
have the strict bound (for x ≥ 2657)
|π(x) − li0(x)| < 1 √
x ln x,
8π
where li0(x) is defined in Exercise 1.36. Show via computations that none of
the data in Table 1.1 violates the Riemann hypothesis!
By direct computation and the fact that li (x) < li0(x) < li (x) +1.05,
prove the assertion in the text that assuming the Riemann hypothesis,
|π(x) − li (x)| < √ x ln x for x ≥ 2.01. (1.48)
It follows from the discussion in connection to (1.25) that (1.48) is equivalent
to the Riemann hypothesis. Note too that (1.48) is an elementary assertion,
which to understand one needs to know only what a prime is, the natural
logarithm, and integrals. Thus, (1.48) may be considered as a formulation of
the Riemann hypothesis that could be presented in, say, a calculus course.
1.38. With ψ(x) defined as in (1.22), it was shown in [Schoenfeld 1976] that
theRiemannhypothesisimpliesthat
|ψ(x) − x| < 1 √ 2
x ln x for x ≥ 73.2.
8π
By direct computation show that on assumption of the Riemann hypothesis,
|ψ(x) − x| < √ x ln 2 x for x ≥ 3.