Prime Numbers

440 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
of the RH, but because of a strong interdisciplinary flavor in what follows, the
description belongs here just as well.
Consider these RH equivalences as research directions, primarily computational
but always potentially theoretical:
(1) There is an older, Riesz condition [Titchmarsh 1986, Section 14.32] that
is equivalent to the RH, namely,
∞
n=1
(−x) n
= O x
ζ(2n)(n − 1)! 1/4+ε
.
Note the interesting feature that only integer arguments of ζ appear.
One question is this: Can there be any value whatsoever in numerical
evaluations of the sum? If there be any value at all, methods for socalled
“recycled” evaluations of ζ come into play. These are techniques
for evaluating huge sets of ζ values having the respective arguments in
arithmetic progression [Borwein et al. 2000].
(2) The work of [Balazard et al. 1999] proves that
I =
ln |ζ(s)|
|s| 2 ds =2π
Re(ρ)>1/2
ln ρ
,
1 − ρ
where the line integral is carried out over the critical line, and ρ denotes
any zero in the critical strip, but to the right of the critical line as indicated,
counting multiplicity. Thus the simple statement “I = 0” is equivalent to
the RH. One task is to plot the behavior of I(T ), which is the integral I
restricted to Im(s) ∈ [−T,T], and look for evident convergence I(T ) → 0,
possibly giving a decay estimate. Another question mixes theory and
computation: If there is a single errant zero ρ = σ + it with σ > 1/2
(and its natural reflections), and if the integral is numerically computed
to some height T and with some appropriate precision, what, if anything,
can be said about the placement of that single zero? A challenging question
is: Even if the RH is true, what is a valid positive α such that
I(T )=O(T −α )?
It has been conjectured [Borwein et al. 2000] that α = 2 is admissible.
(3) Some new equivalences of the RH involve the standard function
ξ(s) = 1
2 s(s − 1)π−s/2 Γ(s/2)ζ(s).
The tantalizing result in [Pustyl’nikov 1999] says that a condition
applicable at a single point s =1/2 as
d n ξ
ds n
1
> 0,
2