Prime Numbers

1.4 Analytic number theory 39
in the ongoing RH verification is S. Wedeniwski, who maintains a “zetagrid”
distributed project [Wedeniwski 2004].
Another result along similar lines is the recent settling of the “Mertens
conjecture,” that
|M(x)| < √ x. (1.26)
Alas, the conjecture turns out to be ill-fated. An earlier conjecture that the
right-hand side could be replaced by 1√
2 x was first disproved in 1963 by
Neubauer; later, H. Cohen found a minimal (least x) violation in the form
M(7725038629) = 43947.
But the Mertens conjecture (1.26) was finally demolished when it was shown
in [Odlyzko and te Riele 1985] that
lim sup x −1/2 M(x) > 1.06,
lim inf x −1/2 M(x) < −1.009.
It has been shown by Pintz that for some x less than 101065 the ratio M(x)/ √ x
is greater than 1 [Ribenboim 1996]. Incidentally, it is known from statistical
theory that the summatory function m(x) =
n≤x tn of a random walk (with
tn = ±1, randomly and independently) enjoys (with probability 1) the relation
lim sup
m(x)
(x/2) ln ln x =1,
so that on any notion of sufficient “randomness” of the Möbius µ function
M(x)/ √ x would be expected to be unbounded.
Yet another numerical application of the Riemann zeta function is in the
assessment of the prime-counting function π(x) for particular, hopefully large
x. We address this computational problem later, in Section 3.7.2.
Analytic number theory is rife with big-O estimates. To the computationalist,
every such estimate raises a question: What constant can stand in place
of the big-O and in what range is the resulting inequality true? For example,
it follows from a sharp form of the prime number theorem that for sufficiently
large n, then-th prime exceeds n ln n. It is not hard to see that this is true
for small n as well. Is it always true? To answer the question, one has to
go through the analytic proof and put flesh on the various O-constants that
appear, so as to get a grip on the “sufficiently large” aspect of the claim. In a
wonderful manifestation of this type of analysis, [Rosser 1939] indeed showed
that the n-th prime is always larger than n ln n. Later, in joint work with
Schoenfeld, many more explicit estimates involving primes were established.
These collective investigations continue to be an interesting and extremely
useful branch of computational analytic number theory.
1.4.3 Dirichlet L-functions
One can “twist” the Riemann zeta function by a Dirichlet character. To
explain what this cryptic statement means, we begin at the end and explain
what is a Dirichlet character.