Prime Numbers

1.5 Exercises 69
1.64. Perform computations that connect the distribution of primes with the
Riemann critical zeros by way of the ψ function defined in (1.22). Starting
with the classical exact relation (1.23), obtain a numerical table of the first 2K
critical zeros (K of them having positive imaginary part), and evaluate the
resulting numerical approximation to ψ(x) for, say, noninteger x ∈ (2, 1000).
As a check on your computations, you should find, for K = 200 zeros and
denoting by ψ (K) the approximation obtained via said 2K zeros, the amazing
fact that
ψ(x) (200)
− ψ (x) < 5
throughout the possible x values. This means—heuristically speaking—
that the first 200 critical zeros and their conjugates determine the prime
occurrences in (2, 1000) “up to a handful,” if you will. Furthermore, a plot of
the error vs. x is nicely noisy around zero, so the approximation is quite good
in some sense of average. Try to answer this question: For a given range on x,
about how many critical zeros are required to effect an approximation as good
as |ψ − ψ (K) | < 1 across the entire range? And here is another computational
question: How numerically good is the approximation (based on the Riemann
hypothesis)
ψ(x) =x +2 √ x
t
sin(t ln x)
t
+ O √ x ,
with t running over the imaginary parts of the critical zeros [Ellison and
Ellison 1985]? For an analogous analytic approach to actual prime-counting,
see Section 3.7 and especially Exercise 3.50.
1.65. This, like Exercise 1.64, also requires a database of critical zeros of the
Riemann zeta function. There exist some useful tests of any computational
scheme attendant on the critical line, and here is one such test. It is a
consequence of the Riemann hypothesis that we would have an exact relation
(see [Bach and Shallit 1996, p. 214])
ρ
1
=2+γ − ln(4π),
|ρ| 2
where ρ runs over all the zeros on the critical line. Verify this relation
numerically, to as much accuracy as possible, by:
(1) Performing the sum for all zeros ρ =1/2+it for |t| ≤T , some T of choice.
(2) Performing such a sum for |t| ≤ T but appending an estimate of
the remaining, infinite, tail of the sum, using known formulae for the
approximate distribution of zeros [Edwards 1974], [Titchmarsh 1986], [Ivić
1985].
Note in this connection Exercises 1.61 (for actual calculation of ζ values)
and 8.34 (for more computations relating to the Riemann hypothesis).
1.66. There are attractive analyses possible for some of the simpler
exponential sums. Often enough, estimates—particularly upper bounds—on