MATH 361: NUMBER THEORY — NINTH LECTURE

1. Algebraic numbers and algebraic integers

√
We like numbers such as i and ω = ζ3 = e2πi/3 and ϕ = (1 + 5)/2 and so
on. To think about such numbers in a structured way is to think of them not as
radicals, but as roots.
Definition 1.1. A complex number α is an algebraic number if α satisfies some
monic polynomial with rational coefficients,
p(α) = 0, p(x) = xn + c1 xn−1 + · · · + cn , c1 , . . . , cn ∈ Q.
Every rational number r is algebraic because it satisfies the polynomial x − r,
but not every algebraic number is rational,√ as shown by the examples given just
before the definition. For example, (1 + 5)/2 satisfies the polynomial p(x) =
x2 − x − 1. Every complex number expressible over Q in radicals is algebraic, but
not conversely.
The algebraic numbers form a field, denoted Q. This is shown as follows.
Theorem 1.2. Let α be a complex number. The following conditions on α are
equivalent:
(1) α is an algebraic number, i.e., α ∈ Q.
(2) The ring Q[α] is a finite-dimensional vector space over Q.
(3) α belongs to a ring R in C that is a finite-dimensional vector space over Q.
Proof. (1) =⇒ (2): Let α satisfy the monic polynomial p(x) ∈ Q[x], and let
n = deg(p). For any nonnegative integer m, the division algorithm in Q[x] gives
xm = q(x)p(x) + r(x), deg(r) < n or r = 0.
Thus, because p(α) = 0,
αm = r(α) ∈ Q ⊕ Qα ⊕ · · · ⊕ Qαn−1 .
Because Q[α] is generated over Q as a vector space by the nonnegative powers of α,
this shows that Q[α] is generated by the finite set {1, α, · · · , αn−1 }.
(2) =⇒ (3) is immediate: let R = Q[α].
(3) =⇒ (1): Let the ring R have basis v1 , . . . , vn as a vector space over Q. Mul-
tiplying each generator by α gives a rational linear combination of the generators,
Xn
αvi = cij vj , i = 1, · · · , n.
j=1
n×n
That is, letting M = [cij ] ∈ Q ,
   
v1 v1
 ..   .. 
α .  = M  . .
vn vn
This shows that α is an eigenvalue of M , and so it satisfies the characteristic
polynomial of M , a monic polynomial with rational coefficients. 
1
2 MATH 361: NUMBER THEORY — NINTH LECTURE

The implications (1) =⇒ (2) =⇒ (3) in the theorem are essentially trivial. The
one idea in the theorem is the argument that (3) =⇒ (1) in consequence of α being
an eigenvalue. Here the ring structure and the vector space structure of R interact.
For example, if we take α = ω = ζ3 = e2πi/3 and R = Q[α] then multiplication by α
takes 1 to α and α to α2 = −1 − α, so the matrix in the proof that (3) =⇒ (1) is
M = −10 −11 , whose characteristic polynomial det(xI − M ) = x2 + x + 1 is indeed
the characteristic polynomial of α.
Condition (3) in the theorem easily proves
Corollary 1.3. The algebraic numbers Q form a field.
Proof. Let α and β be algebraic numbers. Then the rings Q[α] and Q[β] have
respective bases
{αi : 0 ≤ i < m} and {β j : 0 ≤ j < n}
as vector spaces over Q. Let
R = Q[α, β],
spanned as a vector space over Q by the set
{αi β j : 0 ≤ i < m, 0 ≤ j < n}.
Then α + β and αβ belong to R, making them algebraic numbers by condition (3)
of the theorem. If α 6= 0 then its polynomial p(x) can be taken to have a nonzero
constant term cn after dividing through by its lowest power of x. The relations
α | cn − p(α) in Q[α] and p(α) = 0 give α | cn in Q[α], so that
α−1 = (1/cn ) · (cn /α) ∈ Q[α],
making α−1 an algebraic number by condition (3) as well. 
The end of the proof just given, showing that the inverse of a nonzero algebraic
number α is again algebraic, can be written more formulaically as follows. The
relation p(α) = 0 is
αn + c1 αn−1 + · · · + cn−1 α + cn = 0, cn 6= 0,
or
α(αn−1 + c1 αn−2 + · · · + cn−1 ) = −cn ,
and so
α−1 = −c−1
n (α
n−1
+ c1 αn−2 + · · · + cn−1 ).
If α and β are algebraic numbers satisfying the monic rational polynomials p(x)
and q(x) then the proofs of Corollary 1.3 and of (3) =⇒ (1) in Theorem 1.2
combine to produce the polynomials
√ satisfied by α + β and αβ and 1/α if α 6= 0.
For example, let α = i and β = 2. Then
√ √ √
Q[i, 2] = Q ⊕ Qi ⊕ Q 2 ⊕ Qi 2.
Compute that
    
1 0 1 1 0 1
√  i   −1
√   0   √i  .
0 1   
(i + 2) 
 2 =  2
√ 0 0 1   √2 
i 2 0 2 −1 0 i 2
√
Thus i + 2 satisfies the characteristic polynomial of the matrix in the display.
 MATH 361: NUMBER THEORY — NINTH LECTURE 3

The theory of resultants provides a general algorithm to find such polynomi-

als. The idea is that given any field k, and given any two nonzero polynomials
f (T ), g(T ) ∈ k[T ], their resultant
R(f (T ), g(T )) ∈ k
is zero if and only if f and g share a root that is algebraic over k. That is:
The condition R(f (T ), g(T )) = 0 eliminates the variable T from
the simultaneous equations f (T ) = 0, g(T ) = 0.
Now, suppose that the algebraic numbers α and β respectively satisfy the polyno-
mials f (T ) and g(U ) over Q. Then the condition
R(f (T ), R(g(U ), T + U − V )) = 0
first eliminates U from the simultaneous conditions g(U ) = 0, V = T + U , leaving a
polynomial condition h(T, V ) = 0, and then it eliminates T from the simultaneous
conditions f (T ) = 0, h(T, V ) = 0, leaving a polynomial k(V ) over Q having α + β
as a root. Almost identically, the condition
R(f (T ), R(g(U ), T U − V )) = 0
is a polynomial condition k(V ) over Q having αβ as a root.
One can now consider complex numbers α satisfying monic polynomials with
coefficients in Q. But in fact Q is algebraically closed, meaning that any such α
is already in Q. The proof again uses condition (3) in Theorem 1.2.

Corollary 1.4. The field Q of algebraic numbers is algebraically closed.

Proof. (Sketch.) Consider a monic polynomial whose coefficients are algebraic num-
bers,
xn + c1 xn−1 + · · · + cn , ci ∈ Q,
and let α be one of its roots. Because each ring Q[ci ] is a finite-dimensional vector
space over Q, so is the ring
Ro = Q[c1 , . . . , cn ].
Let
R = Ro [α].
If {vi : 1 ≤ i ≤ m} is a basis for Ro over Q then
{vi αj : 1 ≤ i ≤ m, 0 ≤ j < n}
is a spanning set for R as a vector space over Q. (This set is not necessarily a basis
because α might satisfy a polynomial of degree lower than n.) Now condition (3)
of the theorem shows that α ∈ Q. 

A loose analogy holds here:

Just as the finite-cover notion of compactness clarifies analytic
phenomena by translating them into topological terms, so finite
generation clarifies algebraic phenomena by translating them into
structural terms.
4 MATH 361: NUMBER THEORY — NINTH LECTURE

The slogan for the proof that the field of algebraic numbers is algebraically closed
is finitely generated over finitely generated is finitely generated .
The ring of integers Z in the rational number field Q has a natural analogue
in the field of algebraic numbers Q. To begin discussing this situation, note that
any algebraic number satisfies a unique monic polynomial of lowest degree, because
subtracting two distinct monic polynomials of the same degree gives a nonzero
polynomial of lower degree, which can be rescaled to be monic. The unique monic
polynomial of least degree satisfied by an algebra number α is called the minimal
polynomial of α.
Definition 1.5. An algebraic number α is an algebraic integer if its minimal
polynomial has integer coefficients.
The set of algebraic integers is denoted Z. Immediately from the definition,
the algebraic integers in the rational number field Q are the usual integers Z, now
called the rational integers. Also in consequence of the definition, a small exercise
shows that every algebraic number takes the form of an algebraic integer divided
by a rational integer. Note, however, that the algebraic numbers
√ √
−1 + i 3 1+ 5
ω= and ϕ =
2 2
are algebraic integers despite “having denominators”—indeed, they satisfy the poly-
nomials x2 + x + 1 and x2 − x − 1 respectively. Similarly to Theorem 1.2 and its
corollaries,
Theorem 1.6. Let α be a complex number. The following conditions on α are
equivalent:
(1) α is an algebraic integer, i.e., α ∈ Z,
(2) The ring Z[α] is finitely generated as an Abelian group,
(3) α belongs to a ring R in C that is finitely generated as an Abelian group.
Corollary 1.7. The algebraic integers Z form a ring.
Corollary 1.8. The algebraic integers form an integrally closed ring, meaning
that every monic polynomial with coefficients in Z factors down to linear terms
over Z, i.e., its roots lie in Z.
A vector space over Q is a Q-module, and an Abelian group is a Z-module; so
conditions (3) in Theorems 1.2 and 1.6 can be made uniform, and conformal with
parts (1) and (2) of their theorems, by phrasing them as, “α belongs to a ring R
in C that is finitely generated as a Q-module,” and “. . . as a Z-module.”

2. Quadratic Reciprocity Revisited

We work in the ring Z of algebraic integers, remembering at the end that an
algebraic integer congruence between two rational integers is in fact a rational
integer congruence. (Proof: If
a, b ∈ Z and a = b mod nZ,
then
b−a
∈ Q ∩ Z = Z,
n
 MATH 361: NUMBER THEORY — NINTH LECTURE 5

so that
a = b mod nZ,
as desired.)
Let p be an odd prime. To evaluate the Legendre symbol (2/p), introduce not
the square root of unity but the eighth root of unity,
ζ = ζ8 = e2πi/8 ∈ Z.
Because ζ 4 = −1, also ζ 2 + ζ −2 = 0, and thus −1 2
√ (ζ + ζ ) = 2. (This equality is
±1
also clear from the fact that ζ = (1 ± i)/ 2, but the given derivaton uses only
the fact that ζ is a primitive eighth root of unity, not its description as a complex
number.) Further, a small calculation shows that working modulo pZ,
(
ζ + ζ −1 if p = ±1 mod 8,
(ζ + ζ −1 )p =
−(ζ + ζ −1 ) if p = ±3 mod 8.
2
= (ζ + ζ −1 )(−1)(p −1)/8
.
Let
τ = ζ + ζ −1 ,
and compute τ p+1 in two different ways. First, using Euler’s law at the last step,
 
p+1 2 2 (p−1)/2 (p−1)/2 pZ 2
τ = τ (τ ) =2·2 ≡ 2 .
p
And second, quoting the small calculation,
pZ 2
−1)/8 2
−1)/8
τ p+1 = τ · τ p ≡ τ 2 (−1)(p = 2(−1)(p .
Thus we have a congruence in Z,
 
2 2
2 = 2(−1)(p −1)/8 mod pZ,
p
but because both quantities are rational integers we may view the congruence as
being set in Z. Because p is odd, we may cancel the 2’s,
 
2 2
= (−1)(p −1)/8 mod pZ,
p
and again because p is odd and because the integers on each side of the congruence
are ±1, the integers must be equal,
 
2 2
= (−1)(p −1)/8 .
p

The proof of the main quadratic reciprocity law is similar. Let p and q be distinct
odd primes. (In this argument, p and q will play roles respectively analogous to
those played by 2 and p a moment ago.) Introduce the pth root of unity,
ζ = ζp = e2πi/p .
The finite geometric sum formula gives
p−1
(
X
at p−1 if t = 0 mod p,
ζ =
a=1
−1 6 0 mod p.
if t =
6 MATH 361: NUMBER THEORY — NINTH LECTURE

Define the Gauss sum

p−1  
X t
τ= ζ t.
t=1
p
(Yes, the Gauss sum seems to come out of nowhere. In fact it is a very easy case
of a Lagrange resolvent.) Compute that the Gauss sum lets us express p in terms
of pth roots of unity,
Xs t  X  s   su  X u X
τ2 = ζ s+t = ζ s(1+u) = ζ s(1+u)
s,t
p p s,u
p p u
p s
X  u   −1  X  u   −1 
=− + (p − 1) = − + p
p p u
p p
u6=−1
= p∗ , where p∗ denotes whichever of ±p is 1 mod 4.
Now similarly to above, we compute τ q+1 in two ways. First, by Euler’s Law,
 ∗
q+1 2 2 (q−1)/2 ∗ ∗ (q−1)/2 qZ ∗ p
τ = τ (τ ) = p (p ) ≡ p .
q
And second, noting for the second equality to follow that (t/p)q = (tq 2 /p) =
(tq/p)(q/p) with (q/p) independent of t,
X  t q X  qt   
q
 
q
 
q
q+1 q qZ qt qt 2 ∗
τ =τ ·τ ≡ τ ζ =τ ζ · =τ =p .
t
p t
p p p p

Thus we have a congruence in Z,

 ∗  
∗ p ∗ q
p =p mod qZ,
q p
but because both quantities are rational integers we may view the congruence as
being set in Z. Because p and q are distinct, we may cancel the p∗ ’s,
 ∗  
p q
= mod qZ,
q p
and because q is odd and because the integers on each side of the congruence are ±1,
the integers must be equal,
 ∗  
p q
= .
q p

3. Sketch of a Modern Proof of Quadratic Reciprocity

Let p be an odd prime, let ζ = e2πi/p , and consider the cyclotomic number field
K = Q(ζ).
Its Galois group
G = Gal(K/Q)
×
is naturally isomorphic to (Z/pZ) , the automorphism that takes ζ to ζ m mapping
to the residue class m mod p for m ∈ {1, . . . , p − 1}.
Because the Galois group is cyclic of even order, the cyclotomic field K has a
unique quadratic subfield. To describe this field, let p∗ = (−1)(p−1)/2 p; thus p∗ is
 MATH 361: NUMBER THEORY — NINTH LECTURE 7

whichever of ±p equals 1 mod 4. We know that p∗ is a square in K, the square of

the Gauss sum τ . Consequently, the unique quadratic subfield of K is
√
F = Q( p∗ ).
Its Galois group
Q = Gal(F/Q)
√
is naturally
√ ∗ isomorphic to {±1}, the nontrivial automorphism that takes p∗
to − p mapping to −1. Summarizing so far, we have a commutative diagram
in which the horizontal arrows are isomorphisms, so that the left vertical arrow
(restriction of automorphisms) gives rise to the right vertical arrow,

G / (Z/pZ)×

 
Q / {±1}.

Any odd prime q 6= p is unramified in K, and so it has a unique Frobenius

automorphism,
Frobq,K ∈ G,
whose action on the integers of K, written in exponential notation, is characterized
by the condition
xFrobq,K = xq mod qOK , x ∈ OK .
The Frobenius automorphism has no choice but to be
Frobq,K : ζ 7−→ ζ q .
The odd prime q 6= p is also unramified in F , so that again it has a unique Frobenius
automorphism, this time denoted
Frobq,F ∈ Q,
characterized by the condition
xFrobq,F = xq mod qOF , x ∈ OF ,
∗ (q−1)/2
and (because (p ) = (p /q) mod q, where (p∗ /q) is the Legendre symbol)
∗

working out to
√ √
Frobq,F : p∗ 7−→ (p∗ /q) p∗ .
Finally, Frobq,F is the restriction of Frobq,K to F . So, in the commutative diagram
we have
 / q mod p
Frobq,K _
_

 
Frobq,F  / (p∗ /q).
The right vertical arrow shows that:
As a function of q, (p∗ /q) depends only on q mod p.
This is quadratic reciprocity. A less conceptual but more concrete variant of the
punchline is that the map down the right side is q mod p 7−→ (q/p), and so the
commutative diagram shows that (p∗ /q) = (q/p).
8 MATH 361: NUMBER THEORY — NINTH LECTURE

4. The Sign of the Quadratic Gauss Sum

Let p be an odd prime and let p∗ = (−1)(p−1)/2 p; thus p∗ is whichever of ±p
equals 1 mod 4. Let ζ = e2πi/p and let τ denote the quadratic Gauss sum modulo p,
p−1
X
τ= (t/p)ζ t .
t=1
2 ∗
We know that τ = p , so that
( √
± p if p = 1 (mod 4),
τ= √
±i p if p = 3 (mod 4).

Ireland and Rosen narrate Gauss’s original demonstration that in both cases the
sign is “+”, and their exposition is rewritten here. However, the sign is readily
found by a Poisson summation argument, to be given in the next section, so the
reader should feel free to skip to there absent the desire to see a more elementary
argument.
√
The proof first establishes that a certain product equals p if p = 1 mod 4 and
√
equals i p if p = 3 mod 4, and then it establishes that this product also equals the
Gauss sum.
It is elementary that
p−1 p−1 (p−1)/2
X Y Y
Xj = (X − ζ j ) = (X − ζ j )(X − ζ −j ).
j=0 j=1 j=1

But the product as written is overspecific in that the exponents of ζ need only to
vary through any set of nonzero residue classes modulo p. The residue system that
will help us here is the length-(p − 1) arithmetic progression of 2 (mod 4) numbers
symmetrized about 0,
p−1
±(4 · 1 − 2), ±(4 · 2 − 2), ±(4 · 3 − 2), · · · , ±(4 · 2 − 2).
Thus
p−1 (p−1)/2
X Y
Xj = (X − ζ 4j−2 )(X − ζ −(4j−2) ).
j=0 j=1

Substitute X = 1 to get
(p−1)/2
Y
p= (1 − ζ 4j−2 )(1 − ζ −(4j−2) )
j=1
(p−1)/2
Y
= (ζ −(2j−1) − ζ 2j−1 )(ζ 2j−1 − ζ −(2j−1) ),
j=1

and so multiplying by (−1)(p−1)/2 gives

(p−1)/2
Y
p∗ = (ζ 2j−1 − ζ −(2j−1) )2 .
j=1
 MATH 361: NUMBER THEORY — NINTH LECTURE 9

It follows that
(p−1)/2
( √
Y ± p if p = 1 (mod 4),
(ζ 2j−1 − ζ −(2j−1) ) = √
j=1
±i p if p = 3 (mod 4).

The jth multiplicand is

ζ 2j−1 − ζ −(2j−1) = 2i sin(2π(2j − 1)/p),
and the sine is positive for 0 < 2(2j − 1)/p < 1, or 1/2 < j < p/4 + 1/2, or
1 ≤ j < p/4 + 1/2; and similarly the sine is negative for p/4 + 1/2 < j ≤ (p − 1)/2.
If p = 1 (mod 4) then the sine is positive for j = 1, · · · , (p − 1)/4, and so
(p−1)/2
Y
(ζ 2j−1 − ζ −(2j−1) ) = (positive number) × i(p−1)/2 (−1)(p−1)/2−(p−1)/4
j=1

= (positive number) × (−1)(p−1)/4 (−1)(p−1)/4 ,

a positive number. If p = 3 (mod 4) then the sine is positive for j = 1, · · · , (p+1)/4,
and so
(p−1)/2
Y
(ζ 2j−1 − ζ −(2j−1) ) = (positive number) × i(p−1)/2 (−1)(p−1)/2−(p+1)/4
j=1

= (positive number) × i(−1)(p−3)/4 (−1)(p−3)/4 ,

a positive multiple of i. Thus both “±” signs are positive,
(p−1)/2
(√
Y
2j−1 −(2j−1) p if p = 1 (mod 4),
(ζ −ζ )= √
j=1
i p if p = 3 (mod 4).

To complete the argument, we need to show that the Gauss sum τ equals the
Q(p−1)/2 2j−1
product j=1 (ζ − ζ −(2j−1) ) rather than its negative.
Let
(p−1)/2
Y
τ =ε (ζ 2j−1 − ζ −(2j−1) ),
j=1

where we know that ε = ±1 and we want to show that ε = 1. Consider the

polynomial
p−1 (p−1)/2
X Y
t
f (X) = (t/p)X − ε (X 2j−1 − X p−(2j−1) ).
t=1 j=1

Then f (1) = 0 and f (ζ) = 0. So f (X) is divisible by X p − 1,

p−1 (p−1)/2
X Y
(t/p)X t − ε (X 2j−1 − X p−(2j−1) ) = (X p − 1)g(X).
t=1 j=1
z
Replace X by e to get
p−1 (p−1)/2
X Y
(1) (t/p)etz − ε (e(2j−1)z − e(p−(2j−1))z ) = (epz − 1)g(ez ).
t=1 j=1
10 MATH 361: NUMBER THEORY — NINTH LECTURE

On the left side of (1), each multiplicand has constant term 0, so that the lowest
exponent of z in the product is (p − 1)/2, and each multiplicand has linear term
(4j − p − 2)z. Thus the overall coefficient of z (p−1)/2 on the left side of (1) is
Pp−1 (p−1)/2 (p−1)/2
t=1 (t/p)t
Y
−ε (4j − p − 2).
((p − 1)/2)! j=1

On the right side of (1), each coefficient of the power series expansion
∞
X pn n
epz − 1 = z
n=1
n!
has more powers of p in its numerator than in its denominator. Thus the coefficient
of z (p−1)/2 on the left side of (1) is 0 modulo p, and so after clearing a denominator
we have
p−1 (p−1)/2
X p Y
(t/p)t(p−1)/2 ≡ ε((p − 1)/2)! (4j − 2).
t=1 j=1
Working modulo p, and quoting Euler’s Law and then Fermat’s Little Theorem,
the left side is
p−1
X p−1
X p−1
X p−1
X
(t/p)t(p−1)/2 = t(p−1)/2 t(p−1)/2 = tp−1 = 1 = −1,
t=1 t=1 t=1 t=1
and the right side is
(p−1)/2 (p−1)/2
Y Y
ε((p − 1)/2)! (4j − 2) = ε((p − 1)/2)!2(p−1)/2 (2j − 1)
j=1 j=1
= ε(2 · 4 · · · (p − 1))(1 · 3 · · · (p − 2))
= ε(p − 1)!
= −ε by Wilson’s Theorem.
So ε = 1 and the argument is complete.

5. The Sign of the Quadratic Gauss Sum by Fourier Analysis

Let p be an odd prime and let p∗ = (−1)(p−1)/2 p; thus p∗ is whichever of ±p
equals 1 mod 4. Let ep (x) = e2πix/p for x ∈ R, and let τ denote the quadratic
Gauss sum modulo p,
p−1
X
τ= (t/p)ep (t).
t=1
We know that τ 2 = p∗ , so that
( √
± p if p = 1 (mod 4),
τ= √
±i p if p = 3 (mod 4).
We show again, using Fourier analysis this time, that in both cases the sign is “+”.
Begin with the observations that (letting R stand for residue and N for non-
residue)
X X
τ= ep (R) − ep (N),
R N
 MATH 361: NUMBER THEORY — NINTH LECTURE 11

while by the finite geometric sum formula

X X
0=1+ ep (R) + ep (N),
R N

so that adding the previous two displays shows that the Gauss sum is
p−1 p−1
2πin2 /p 2
X X X
τ =1+2 ep (R) = e = e2πip(n/p) .
R n=0 n=0

We will evaluate the more general sum associated to any positive integer N ,
N −1
X 2
SN = e2πiN (n/N ) , N ∈ Z>0 .
n=0

The result, encompassing the sign of the Gauss sum, will be that

1 + i if N
 =0 mod 4
√

1 if N =1 mod 4
SN = N ·
0
 if N =2 mod 4

i if N =3 mod 4.


1.0

0.5

0.2 0.4 0.6 0.8 1.0

-0.5

-1.0

2
Figure 1. Real and imaginary parts of f (x) = e2πiN x for N = 3

The summands of SN are values of the function

2
f : [0, 1] −→ C, f (x) = e2πiN x
(see Figure 1). As will be explained below, the Fourier series of f converges point-
wise to f even at 0 because f (1) = f (0) and f is differentiable from the right at 0
and from the left at 1. Thus
N
X −1 N
X −1 X Z 1
SN = f (n/N ) = f (x)e2πikx dx · e−2πikn/N
n=0 n=0 k∈Z 0

XZ 1 N
X −1
= f (x)e 2πikx
dx · e−2πikn/N .
k∈Z 0 n=0
12 MATH 361: NUMBER THEORY — NINTH LECTURE

The inner sum is N if k ∈ N Z and 0 otherwise, so now

XZ 1 XZ 1 2
SN = N f (x)e2πiN kx dx = N e2πiN (x +kx) dx.
k∈Z 0 k∈Z 0

To analyze the kth summand, complete the square in the exponent, and then note
that e−2πi/4 = i−1 and that k 2 is 0 mod 4 for k even and is 1 mod 4 for k odd,
Z 1 Z 1
2 2 2
e2πiN (x +kx) dx = e−2πiN k /4 e2πiN (x+k/2) dx
0 0
Z k/2+1
2 2
= i−N k e2πiN x dx
k/2
( )Z
1 if k is even k/2+1
2
= −N
e2πiN x dx.
i if k is odd k/2

As k varies through the even integers, the last integral in the previous display runs
over R, and similarly for the odd integers. Thus summing over the last expression
in the previous display gives
−N
Z
2πiN x2
√ −N
Z
2
SN = N (1 + i ) e dx = N (1 + i ) e2πix dx.
R R
2πix2
R
Note that the last integral I = R e dx in the previous display is independent
of N . In a moment we will show that I converges, perhaps surprisingly because its
√ does not go to 0 as x goes to ±∞. Granting the convergence, the formula
integrand
SN = N (1 + i−N )I for N = 1 is 1 = (1 − i)I, giving I = (1 + i)/2. Thus the
general value of SN is
√ (1 + i−N )(1 + i)
SN = N .
2
The casewise formula for SN follows immediately.
2
The integral I = R e2πix dx converges because its integrand oscillates ever
R

faster with unit amplitude as |x| grows, making its value stabilize. To establish the
convergence analytically, make a change of variable and then integrate by parts:
for 0 < L ≤ M ,
Z M Z 2
2 1 M −1/2 2πiu
e2πix dx = u e du where u = x2
L 2 L2
Z 2
M2 1 M −3/2 2πiu
 
1 −1/2 2πiu
= u e + u e du ,
4πi L2 2 L2
which is asymptotically of order L−1 .
Finally, to show the pointwise convergence of the Fourier series of f to f at 0,
replace f by f − f (0) and compute the M th partial sum of the Fourier series at 0,
XM Z 1 Z 1 M
X
f (x)e2πikx dx = f (x) e2πikx dx
k=−M 0 0 k=−M
Z 1
f (x)
= (e2πi(M +1)x − e−2πiM x ) dx.
0 e2πix − 1
 MATH 361: NUMBER THEORY — NINTH LECTURE 13

In the integrand, the term

f (x) f (x) x
= · 2πix
e2πix − 1 x e −1
extends continuously at x = 0 to the right derivative f 0 (0) times the reciprocal
derivative 1/(2πi) of e2πix at 0, and similarly it extends continuously at x = 1.
Thus the integral goes to 0 as M grows, by the Riemann–Lebesgue Lemma.