Prime Numbers

1.5 Exercises 73
(1) Take q = 2 and explain why the above estimate on EN is obvious for
a =0, 1.
(2) Let q = 3, and for a =1, 2 explain using a vector diagram in the complex
plane how the above estimate works.
(3) Let q = 4, and note that for some a values the right-hand side of the above
estimate is actually zero. In such cases, use an error estimate (such as the
conditional result (1.32)) to give sharp, nonzero estimates on EN(a/4) for
a =1, 3.
These theoretical examples reveal the basic behavior of the exponential sum
for small q.
For a computational foray, test numerically the behavior of EN by way of
the following steps:
(1) Choose N =10 5 , q = 31, and by direct summation over primes p ≤ N,
create a table of E values for a ∈ [0,q− 1]. (Thus there will be q complex
elements in the table.)
(2) Create a second table of values of π(N) µ(q/g)
ϕ(q/g) ,alsoforeacha∈ [0,q− 1].
(3) Compare, say graphically, the two tables. Though the former table is
“noisy” compared to the latter, there should be fairly good average
agreement. Is the discrepancy between the two tables consistent with
theory?
(4) Explain why the latter table is so smooth (except for a glitch at the
(a = 0)-th element). Finally, explain how the former table can be
constructed via fast Fourier transform (FFT) on a binary signal (i.e.,
a certain signal consisting of only 0’s and 1’s).
Another interesting task is to perform direct numerical integration to verify
(for small cases of N, say) some of the conjectural equivalences of Exercise
1.67.
1.71. Verify the following: There exist precisely 35084 numbers less than
10 100 that are 4-smooth. Prove that for a certain constant c, thenumberof
4-smooth numbers not exceeding x is
ψ(x, 4) ∼ c ln 2 x,
giving the explicit c and also as sharp an error bound on this estimate as you
can. Generalize by showing that for each y ≥ 2 there is a positive number cy
such that
ψ(x, y) ∼ cy ln π(y) x, where y is fixed and x →∞.
1.72. Carry out some numerical experiments to verify the claim after
equation (1.45) that the implicit lower bound is a “good” one.
1.73. Compute by empirical means the approximate probability that a
random integer having 100 decimal digits has all of its prime factors less than