Math 539
Homework #5
due Monday, December 12, 2005 at 3 PM
Note: additional work on past homeworks should also be handed in by Monday, December 12,
2005 at 3 PM. No extensions will be granted after that date and time.
Reality check problems. Not to write up; just ensure that you know how to do them.
I. Let a(n) be a nonnegative arithmetic function, and suppose that u( x) is a function
for which a(n) ≤ u( x) for all n ≤ x. For any function f ( x), let N f ( x) denote the
number of integers n ≤ x for which a(n) > f ( x). Show that
N f ( x) f ( x)k <
∑ a(n)k ≤ x f ( x)k + N f ( x)u( x)k .
n≤ x
II. Given the integral definition of the Euler Gamma-function
Γ( z) =
Z ∞
0
e−t t z
dt
t
(<z > 0),
deduce the formula Γ( z + 1) = zΓ( z). Conclude that Γ( z) can be analytically continued to C \ {0, −1, −2, . . . }. Also conclude that Γ(n + 1) = n! for every integer
n ≥ 0.
Homework problems. To write up and hand in.
I. Consider the following arithmetic functions:
n;
nn ;
n! ;
539π (n) ;
lcm{1, 2, . . . , n} ;
∏p.
p≤n
Label these functions as f 1 (n), . . . , f 6 (n) in such a way that f 1 (n) f 2 (n) · · · f 6 (n). Then, put the consecutive ratios f 2 (n)/ f 1 (n), . . . , f 6 (n)/ f 5 (n) in order of increasing magnitude as well.
II. (a) From the symmetry ξ (s) = ξ (1 − s), derive the following non-symmetric
functional equation for ζ (s):
ζ (1 − s) = 2(2π )−s cos sπ
2 Γ ( s )ζ ( s ) .
(b) Recall that you proved in problem X(c) of Homework #4 that
Bn
n
for every integer n ≥ 2, where Bn is the nth Bernoulli number. Using this
1
fact and part (a) of this problem, derive a formula for ζ (2k) = ∑∞
n=1 n2k for all
k ∈ N. What does this procedure tell us about ζ (2k + 1)?
ζ (1 − n) = −
III. For any positive real number A, prove that there exists a constant C = C ( A) such
that there is always a prime between x and x + Cx/(log x) A . Derive a “souped-up”
version of Bertrand’s postulate: for any ε > 0, there is always a prime between x
and (1 + ε) x when x is sufficiently large in terms of ε.
IV. Legendre tentatively conjectured that
π ( x) ≈
x
.
log x − 1.08 . . .
(a) Show that no matter what the constant A ∈ R equals, we have
π ( x) ∼
x
log x − A
(∗)
as x → ∞.
(b) Show that in fact, the best approximation of the form (∗) is when A = 1. More
precisely, if the function A( x) is defined by
π ( x) =
x
,
log x − A( x)
determine the constant b such that
b
1
A( x) = 1 +
+O
.
log x
log2 x
(c) Bateman and Diamond, p. 207, #8.14
V. Recall that the constant B in the Hadamard product for ξ (s) took two different
forms:
B = − 21 γ − 1 + 12 log 4π = − lim ∑ ρ1 ,
T →∞ ρ
where the sum is over the nontrivial zeros of the Riemann zeta-function within T of
the real axis, that is, over all complex numbers ρ such that 0 < <ρ < 1, |=ρ| ≤ T,
and ζ (ρ) = 0. Assuming this equality, prove that |=ρ| > 6.56 for all such zeros
ρ. You may use a calculator or computer for numerical evaluations. (Hints: group
the zeros with negative imaginary part with their partners with positive imaginary
part, throw most of the zeros away, and find a way to assume that <ρ ≥ 21 .)
1
2
≤ σ ≤ 1,
(
|t|1−σ log |t|, if |t| ≥ 2,
ζ (s) 1/(1 − σ ),
if |t| ≤ 2.
VI. Show that uniformly for
(Hint: for |t| ≥ 2, derive this from the proof of Lemma 8.4 in Bateman and Diamond; for smaller t, use the Laurent expansion of ζ (s) near s = 1.)
k
+ Ok x(log x)2 −2 , where
2k 1
4k
3k
1
2k
ck = k
+ 2 + 3 +··· .
1−
1+
p
p
p
p
(2 − 1)! ∏
p
VII. Let k ∈ N. Prove that ∑n≤x τ (n)k = ck x(log x)2
k −1
VIII. Let C > 1 be any real number, and define SC = {n ∈ N : τ (n) > (log n)C }. Let
SC ( x) = #{n ∈ SC : n ≤ x} be the counting function of SC .
(a) Show that there exists a function D (C ), positive for every C > 1 and satisfying
limC→∞ D (C )/C = ∞, such that SC ( x) C x(log x)− D(C) .
(b) Show that SC ( x) C,δ x1−δ for every δ > 0.
(Hint for both parts: use problem VIII and Reality Check I. Hint for part (a): try
f ( x) = ( 12 log x)C , and choose the best k as a function of C. Hint for part (b): use
τ (n) ε nε , and choose k (and ε) in terms of C (and δ).
Be very careful when you write this up, in terms of when you have log n and
when you have log x, what parameters the -constants depend upon at each step,
and so on; it’s possible to do this perfectly rigorously in a short argument, but it’s
also possible to write something that seems correct but is invalid.)
IX. (a) Write down all Dirichlet characters to the modulus 15; to the modulus 16.
Indicate which of them are primitive characters.
(b) Show that (φ ∗ µ )(q) is the number of primitive characters to the modulus q.
Calculate the number of characters and the number of primitive characters
modulo 10!.
X. For every finite abelian group G, show that there exist infinitely integers n for
which G is isomorphic to a subgroup of (Z/nZ)× .
XI. Show that an integer that is relatively prime to q has φ(q)/q times as many divisors, on average, as a random integer of the same size. (Hint: use a modified
version of ζ (s), rather than Dirichlet L-functions.)
XII. Let q and a be fixed, relatively prime integers. Prove that
∑
n≤ x
n≡ a (mod q)
µ 2 (n) ∼
6 c(q)
x,
π 2 φ(q)
where c(q) = ∏ p|q p/( p + 1). What does this say about the distribution of squarefree numbers that are relatively prime to q?
XIII. (a) Suppose that 1 ≤ a ≤ q and ( a, q) = 1. Prove that
∑
p≤ x
p≡ a (mod q)
1
1 log log x
+
p
a
φ(q)
and that the term 1a can be omitted if a is not a prime. The -constant should
be absolute. (Hint: use partial summation with a lower bound 2q < p and
invoke the Brun-Titchmarsh inequality.)
(b) Prove that ∑n≤x ν2 (φ(n)) = 2x log log x + O( x), where ν2 is defined in problem VII of Homework #1. (Hint: you’ll be interested in sums like
1
,
∑
p
p≤ x
p≡2k +1 (mod 2k+1 )
for which we have asymptotics when k is small and upper bounds when k is
large.)