Prime Numbers

38 Chapter 1 PRIMES!
conjectured. The famous conjecture of H. Cramér asserts that
lim sup(pn+1
− pn)/ ln
n→∞
2 n =1.
A. Granville has raised some doubt on the value of this limsup, suggesting
thatitmaybeasleastaslargeas2e −γ ≈ 1.123. For primes above 100, the
largest known value of (pn+1 − pn)/ ln 2 n is ≈ 1.210 when pn = 113. The
next highest known values of this quotient are ≈ 1.175 when pn = 1327, and
≈ 1.138 when pn = 1693182318746371, this last being a recent discovery of
B. Nyman.
The prime gaps pn+1 − pn, which dramatically showcase the apparent
local randomness of the primes, are on average ∼ ln n; this follows from the
PNT (Theorem 1.1.4). The Cramér–Granville conjecture, mentioned in the
last paragraph, implies that these gaps are infinitely often of magnitude ln 2 n,
and no larger. However, the best that we can currently prove is that pn+1 −pn
is infinitely often at least of magnitude
ln n ln ln n ln ln ln ln n/(lnlnlnn) 2 ,
an old result of P. Erdős and R. Rankin. We can also ask about the minimal
order of pn+1−pn. The twin-prime conjecture implies that (pn+1−pn)/ ln n has
liminf 0, but until very recently the best we knew was the result of H. Maier
that the liminf is at most a constant that is slightly less than 1/4. As we go to
press for this 2nd book edition, a spectacular new result has been announced
by D. Goldston, J. Pintz, and C. Yildirim: Yes, the liminf of (pn+1 − pn)/ ln n
is indeed 0.
1.4.2 Computational successes
The Riemann hypothesis (RH) remains open to this day. However, it became
known after decades of technical development and a great deal of computer
time that the first 1.5 billion zeros in the critical strip (ordered by increasing
positive imaginary part) all lie precisely on the critical line Re(s) =1/2 [van
de Lune et al. 1986]. It is highly intriguing—and such is possible due to a
certain symmetry inherent in the zeta function—that one can numerically
derive rigorous placement of the zeros with arithmetic of finite (yet perhaps
high) precision. This is accomplished via rigorous counts of the number of
zeros to various heights T (that is, the number of zeros σ + it with imaginary
part t ∈ (0,T]), and then an investigation of sign changes of a certain real
function that is zero if and only if zeta is zero on the critical line. If the sign
changes match the count, all of the zeros to that height T are accounted for
in rigorous fashion [Brent 1979].
The current height to which Riemann-critical-zero computations have
been pressed is that in [Gourdon and Sebah 2004], namely the RH is intact up
to the 10 13 -th zero. Gourdon has also calculated 2 billion zeros near t =10 24 .
This advanced work uses a variant of the parallel-zeta method of [Odlyzko
and Schönhage 1988] discussed in Section 3.7.2. Another important pioneer