1 The Law of Quadratic Reciprocity 139

whereCis theFresnel integral

C=

∫∞

−∞

e^2 πit

2

dt.

(This is an important example of an infinite integral which converges, although the
integrand does not tend to zero.) From(∗)we now obtain the formula forG(m,n)in
the statement of the proposition. To determine the value of the constantC,takem=1,
n=3. We obtaini

√

3 =

√

3 C( 1 +i), which simplifies toC=( 1 +i)/2. 

From Proposition 10 withm=1 we obtain

G( 1 ,n)=

n∑− 1

v= 0

e^2 πiv

(^2) /n

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

( 1 +i)

√

n ifn≡ 0 (mod 4),

√

n ifn≡ 1 (mod 4),

0ifn≡ 2 (mod 4),

i

√

n ifn≡ 3 (mod 4).

Ifmandnare both odd, it follows that

G( 1 ,mn)=G( 1 ,m)G( 1 ,n) if eitherm≡1orn≡1mod4,

=−G( 1 ,m)G( 1 ,n) ifm≡n≡3mod4;

i.e.

G( 1 ,mn)=(− 1 )(m−^1 )(n−^1 )/^4 G( 1 ,m)G( 1 ,n).

If, in addition,mandnare relatively prime, thenG(m,n)G(n,m)=G( 1 ,mn),by
Proposition 9. Hence, if the integersm,nare odd, positive and relatively prime, then

G(m,n)G(n,m)=(− 1 )(m−^1 )(n−^1 )/^4 G( 1 ,m)G( 1 ,n).

For any odd, positive relatively prime integersm,n, put

ρ(m,n)=G(m,n)/G( 1 ,n).

Then

ρ( 1 ,n)= 1 ,

ρ(m,n)=ρ(m′,n) ifm≡m′modn,

ρ(m,n)ρ(n,m)=(− 1 )(m−^1 )(n−^1 )/^4.

We claim thatρ(m,n)is just the Jacobi symbol(m/n). This is evident ifm=1 and,
by Proposition 2(i), ifρ(m,n)=(m/n),thenalsoρ(n,m)=(n/m).
Hence if the claim is not true for allm,n,thereisapairm,nwith 1<m<nsuch
that

ρ(m,n)=(m/n),

butρ(μ,v)=(μ/v)for all odd, positive relatively prime integersμ,vwithμ<m.
We can writen=km+rfor some positive integersk,rwithr<m.