Introduction and Definitions
Some estimates
The Proof Sketch
Sketch of Dirichlet’s Theorem on Arithmetic
Progressions
Tom Cuchta
24 February 2012 / Analysis Seminar, Missouri S&T
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Outline
1
Introduction and Definitions
2
Some estimates
3
The Proof Sketch
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Dirichlet’s theorem: Let h and k be relatively prime positive
integers. Then there are infintiely prime numbers in the set
{kn + h : n ∈ Z+ }.
For example, if h = 2 and k = 3, we are focusing on these red
numbers:
{5, 8, 11, 14, 17, 20, 23, 26, . . .},
and if h = 23 and k = 10, we are focusing on these red
numbers:
{33, 56, 79, 102, 125, 148, 171, 194, 217, 240, 263, . . .}.
We will prove prove this theorem by demonstrating
X
p≡h(mod k )
log p
= ∞.
p
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem (Euler):
X1
p
p
=∞
∞
X
Y
1
1
=
, so
s
n
1 − p−s
p
n=1
X
log ζ(s) = −
log(1 − p−s ).
Proof: For <(s) > 1, ζ(s) =
p
Taylor series: − log(1 − x) = x + O(x 2 ) for x near 0; so for
s > 1 near 1,
log(ζ(s)) =
X
X 1
(p−s + O(p−2s )) =
+ O(1),
ps
p
p
since
∞
X 1
X 1 X 1
X
1
≤
≤
.
=
2s
2s
2<(s)
2<(s)
p
p
p
n
p
p
p
n=1
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem (Euler):
X1
p
p
=∞
Proof:Let s → 1+ :
lim+ log(ζ(s)) = lim+
s→1
s→1
X 1
+ O(1),
ps
p
and the result is
X 1
+ O(1)
ps
s→1
p
X1
=
+ O(1),
p
p
∞ = lim+
since ζ(1) is the harmonic series, which diverges. Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Define U(k ) to be the group of integer units modulo k under
multiplication; for example
U(7) = {1, 2, . . . , 6} ' Z7 ,
U(12) = {1, 5, 7, 11} ' Z4 ,
U(20) = {1, 3, 7, 9, 11, 13, 17, 19} ' Z8 .
It is well known that the cardinality of U(k ) is given by the Euler
totient function
φ(k ) =
X
1,
n≤k ;(n,k )=1
where (n, k ) indicates n is relatively prime to k .
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
We denote the Z/qZ to be the integers modulo q.
It is well known that the nth roots of unity form a multiplicative
subgroup of C of cardinality n. We will denote the set of nth
roots of unity as R(n).
A homomorphism χ : U(k ) → R(k ) is called a character mod k .
It is well known that there are φ(k ) characters mod k .
We label the characters mod k as
{χ1 , χ2 , . . . , χφ(k ) },
with χ1 called the principal character, which sends all elements
of U(q) to 1 (all other characters are considered nonprincipal).
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Let Û(k ) denote the group of all φ(k ) characters mod k under
the operation
χa χb (·) = χa (·)χb (·).
Define L2 (U(k )) to be the set of complex-valued functions on
U(k ) with respect to the inner product
< f , g >=
X
1
f (m)g(m).
φ(k )
m∈U(k )
It is well-known that Û(k ) is an orthonormal basis of L2 (U(k )).
Recall that h is relatively prime to k and define the
characteristic function δh
1 : a ≡ h mod k
δh (a) =
.
0 : otherwise
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Now define
δˆh (χ) =< δh ,X
χ>
1
δh (a)χ(a)
= φ(k )
=
a∈G
1
χ(h).
φ(k )
The following representation for δh is well known:
δh (a) =
φk
X
r =1
δˆh (χr )χr (a) =
φ(k )
1 X
χr (h)χr (a).
φ(k )
r =1
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
We now trivially extend all characters χr mod k to
χr : Z/k Z → R(k ) ∪ {0} by the following formula:
χr (a) : a ∈ U(k )
χr (a) =
.
0
: a ∈ Z/k Z \ U(k )
Such an extended character is called a Dirichlet character mod
k , and we label them {χ1 , . . . , χφ(k ) }.
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
A Dirichlet series is a generalization of the Riemann zeta
function. Define a Dirichlet series Lχ by the formula, for
<(s) > 1,
Lχ (s) =
∞
X
χ(n)
n=1
ns
.
Note that ζ(s) = Lχ1 (s).
It turns out that Dirichlet’s theorem boils down to proving that
Lχ (1) 6= 0 for any nonprincipal character χ mod k .
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
The Dirichlet product of f and g is defined to be f ∗ g and has
formula
n
X
(f ∗ g)(n) =
f (d)g
.
d
d|n
Let F be a complex-valued function which is defined on (0, ∞)
with F (x) = 0 for 0 < x < 1. The general convolution of F and
an arithmetic function f is defined to be
x X
(f ◦ F )(x) =
.
f (n)F
n
n≤x
With a little algebra, can show for arithmetical functions f and g,
that
f ◦ (g ◦ F ) = (f ∗ g) ◦ F .
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: If h = f ∗ g, let
H(x) =
X
h(n), F (x) =
n≤x
X
f (n), and G(x) =
n≤x
X
n≤x
Then,
H(x) =
X
n≤x
f (n)G
x n
=
X
g(n)F
n≤x
Dirichlet’s Theorem
x n
.
g(n).
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: H(x) =
X
f (n)G
x n≤x
n
=
X
g(n)F
n≤x
x n
.
:0<x <1
. Then,
:x ≥1
n
X
X
F (x) =
f (n) =
f (n)U
= (f ◦ U)(x),
x
Proof: Define U(x) =
n≤x
0
1
n≤x
and similarly, G = g ◦ U. So,
X
g(n)F
n≤x
X
n≤x
f (n)G
x n
x n
= g ◦ F = g ◦ (f ◦ U) = (g ∗ f ) ◦ U = H,
= f ◦ G = f ◦ (g ◦ U) = (f ∗ g) ◦ U = H.
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: H(x) =
X
f (n)G
n≤x
x n
=
X
g(n)F
n≤x
x n
.
If we set g(n) = 1 for all n, then G(x) = [x]. The theorem yields
the following corollary:
Corollary: If F (x) =
X
f (n), then
n≤x
XX
n≤x d|n
f (d) =
X
(f ∗ g)(n)
n≤x
= H(x) h i
X
x
=
f (n)
n
n≤x
X x =
F
.
n
n≤x
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Euler Summation formula: if f has continuous derivative f 0 on
the interval [y , x], where 0 < y < x, then
X
y <n≤x
Z
f (n) =
y
x
Z x
f (t)dt+ (t−[t])f 0 (t)dt+f (x)([x]−x)−f (y )([y ]−y ).
y
Define the Mangoldt function
log p : n = pm , p prime,m ≥ 1
Λ(n) =
0
: otherwise.
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Corollary:
XX
f (d) =
n≤x d|n
X
f (n)
n≤x
hx i
n
=
X
F
n≤x
x n
.
Theorem: For x ≥ 1 we have
hx i
X
= log[x]!.
Λ(n)
n
n≤x
Proof: Use fundamental theorem of arithmetic to represent any
k
Y
n as n =
piei . Apply the corollary with f (n) = Λ(n):
i=1
X
n≤x
Λ(n)
hx i
n
=
XX
Λ(d) =
n≤x d|n
So
X
n≤x
Λ(n)
" k
X X
n≤x
hx i
n
#
ei log(pi ) =
i=1
= log([x]!).
Dirichlet’s Theorem
X
n≤x
log(n).
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: If x ≥ 2, we have
log([x]!) = x log x + O(x),
and hence
X
Λ(n)
n≤x
hx i
n
= x log x + O(x).
x
t − [t]
dt = O(log x). Take f (t) = log t
t
1
and apply Euler’s summation formula to get
Z
Proof: Realize that
log([x]!) =
X
1<n≤x
Z
log n =
x
Z
log tdt +
1
1
x
t − [t]
dt + ([x] − x) log x
t
= x log x − x + 1 + O(log x)
= x log x + O(x).
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem:
X x p≤x
p
log p = x log x + O(x)
Proof: Since Λ(n) = 0 unless n = pm , we have
∞ ∞ XX
XX
x
x
m
Λ(p
)
=
log p,
m
m
n
p
p
p m=1
p m=1
n≤x
h i
so since pm ≤ x → p ≤ x and pxm = 0 if p > x, we get
X hx i
Λ(n) =
∞ ∞ XX
X x XX
x
x
log
p
=
log p.
log
p
+
m
m
p
p
p
p
p≤x
m=1
p≤x m=2
So,
X hx i
n≤x
n
Λ(n) =
∞ XX
x
log p +
log p
p
pm
X x p≤x
p≤x m=2
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem:
X x p≤x
p
log p = x log x + O(x)
Now the second term in the RHS is O(x):
∞ ∞
XX
X
X
x
x
log
p
≤
log
p
m
p
pm
p≤x m=2
p≤x
m=2
∞ m
X
X
1
=x
log p
p
p≤x
m=2
!
1
X
p2
=x
log p
1 − p1
p≤x
X
1
=x
log p
p(p − 1)
p≤x
∞
X log n
≤x
= O(x)
n(n − 1)
n=1
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem:
X x p≤x
p
log p = x log x + O(x)
Now we have
∞ XX
x
log p +
Λ(n) =
log p
n
p
pm
p≤x m=2
p≤x X x
log p + O(x).
=
p
X x X hx i
n≤x
p≤x
From earlier, we know
X
n≤x
Λ(n)
hx i
n
= x log x + O(x).
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem:
X x p
p≤x
log p = x log x + O(x)
Combining the formulas
 Xh i
X x x


Λ(n) =
log p + O(x)


n
p
n≤x h i
p≤x
X x


Λ(n) = x log x + O(x),


n
n≤x
implies
X x p≤x
p
log p + O(x) = x log x + O(x),
in other words
X x p≤x
p
log p = x log x + O(x).
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Shapiro’s Tauberian Theorem:
Let {a(n)} be a nonnegative
X hx i
sequence such that
a(n) = x log x + O(x), for all x ≥ 1.
n
n≤x
Then for x ≥ 1 we have
X a(n)
n≤x
n
= log x + O(1).
If n is prime, define Λ1 (n) = log p. Otherwise, define Λ1 (n) = 0.
With this notation, we can write the theorem in the previous
slide as
X hx i
Theorem (previous slide):
Λ1 (n) = x log x + O(x).
n
n≤x
Applying Shapiro’s tauberian theorem to this yields
X log p
p≤x
p
=
X Λ1 (n)
n≤x
n
= log x + O(1).
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
To prove Dirichlet’s theorem,
X
p≡h(mod k )
log p
= ∞,
p
it is sufficient to prove
X
p≤x;p≡h(mod k )
log p
1
=
log x + O(1)
p
φ(k )
(∗)
and then take the limit as x → ∞, because that would yield
lim
x→∞
X
p≤x;p≡h(mod k )
1
log p
= lim
log x + O(1) = ∞.
x→∞ φ(k )
p
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Now we can use our representation for δh and write
X
p≤x;p≡h(mod k )
log p
p
=
X δh (p) log p
p
p≤x


=

φ(k )
1
φ(k )
X
χr (h)χr (p) log p
r =1
X
p
p≤x
φ(k )
=
1
φ(k )
X
χr (h)
1
φ(k )
X log p
p≤x
p
p≤x
r =1
=
X χr (p) log p
p
+
φ(k )
X χr (p) log p
1 X
χr (h)
.
φ(k )
p
r =2
Dirichlet’s Theorem
p≤x
Introduction and Definitions
Some estimates
The Proof Sketch
From earlier, we know
X log p
p≤x
p
= log x + O(1),
so the previous slide yields
X
p≤x;p≡h(mod k )
φ(k )
X χr (p) log p
log p
1 X
= log x+
χr (h)
+O(1).
p
φ(k )
p
r =2
p≤x
If we can prove
φ(k )
X χr (p) log p
1 X
χr (h)
φ(k )
p
r =2
p≤x
remains bounded, then Dirichlet’s theorem would follow by
taking the limit as x → ∞.
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
By a technical lemma,
φ(k )
X χr (p) log p
χr (h)
+ O(1)
p
p≤x
r =2


!
φ(k )
∞
X
X
X
χr (n) log(n) 
µ(n)χr (n) 
1
+ O(1),
χr (h)
φ(k )
n
n
1
φ(k )
=
X
r =2
n≤x
n=1
where µ is the Möbius function defined by
(−1)k : e1 = e2 = . . . = ek = 1
µ(p1e1 . . . pkek ) =
0
: otherwise.
So we will show that
∞
X
χr (n) log(n)
n=1
X µ(n)χr (n)
n≤x
n
n
converges and that
= O(1). Then we are done.
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: Let χ is nonprincipal character mod k , f
non-negative with continuous negative derivative for all x ≥ x0 .
Then if y ≥ x ≥ x0 we have
X
χ(n)f (n) = O(f (x)).
x<n≤y
In addition, if lim f 0 (x) = 0, then
x→∞
X
n≤x
χ(n)f (n) =
∞
X
∞
X
χ(n)f (n) converges and
n=1
χ(n)f (n) + O(f (x)).
n=1
So with f (x) = logx x we have f 0 (x) =
∞
X
χr (n) log(n)
converges.
n
1−log x
x2
and thus the sum
n=1
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
By another technical lemma, for r 6= 1,
Lχr (1)
X µ(n)χr (n)
n≤x
n
= O(1),
So if we can always divide by Lχr (1), we are done. This is why
the proof of Dirichlet’s theorem boils down to showing
Lχ (1) 6= 0 for any nonprincipal character χ mod k .
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: For r > 1 and real-valued χr , Lχr (1) 6= 0.
X
X A(n)
√ . Then it is
Define A(n) =
χr (d), and B(x) =
n
n≤x
d|n
known that lim B(x) = ∞ and
x→∞
√
B(x) = 2 xLχr (1) + O(1)
for all x ≥ 1.
√
Therefore, since
√ Lχr (1) is the coefficient of x and
lim 2Lχr (1) x = ∞, we have Lχr (1) 6= 0. x→∞
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Theorem: For r > 1, non-real-valued χr , Lχr (1) 6= 0.
Let N(k ) denote the number of nonprincipal non-real-valued
characters mod k such that Lχr (1) = 0. Note that if Lχr (1) = 0,
then Lχr (1) = 0 as well, so N(k ) is an even number. By a
technical lemma, if Lχr (1) = 0 then
X
p≤x;p≡1(modk )
log p
1 − N(k )
=
log x + O(1).
p
φ(k )
)
So if N(k ) ≥ 2, the coefficient 1−N(k
φ(k ) will become negative.
This is a contradiction, since all terms in the sum in the left
hand side are positive. Thus N(k ) = 0. Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
In the last few slides, we have shown
X
p≤x;p≡h(mod k )
log p
p
φ(k )
X χr (p) log p
χr (h)
+ O(1)
p
p≤x
r =2


!
φ(k )
∞
X
X
X
χ
(n)
log(n)
µ(n)χ
(n)
r
r
1

 + O
= log x + φ(k
χr (h)
)
n
n
= log x +
1
φ(k )
X
= log x +
1
φ(k )
X
n≤x
n=1
r =2
φ(k )
χr (h)
r =2
= log x + O(1).
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
So, taking a limit yields
lim
x→∞
X
p≤x;p≡h(mod k )
log p
=
p
X
p≡h(mod k )
log p
= lim log x+O(1) = ∞.
x→∞
p
Therefore there are infinitely many primes in the arithmetic
progression {kn + h : n ∈ Z}!
Dirichlet’s Theorem
Introduction and Definitions
Some estimates
The Proof Sketch
Thank you for attending!
References
ANONYMOUS ARTICLE. An introduction to analytic number
theory.
Apostol, Tom. Introduction to Analytic Number Theory.
Springer-Verlag 1976.
Dirichlet’s Theorem