Number Theory: An Introduction to Mathematics

1 The Law of Quadratic Reciprocity 137

G(m,n)=

n∑− 1

v= 0

e^2 πiv

(^2) m/n
.
Instead of summing from 0 ton−1 we can just as well sum over any complete set of
representatives of the integers modn:
G(m,n)=

∑

vmodn

e^2 πiv

(^2) m/n
.
Gauss sums have a useful multiplicative property:
Proposition 9If m,n,n′are positive integers, with n and n′relatively prime, then
G(mn′,n)G(mn,n′)=G(m,nn′).
Proof Whenvandv′run through complete sets of representatives of the integers mod
nand modn′respectively,μ=vn′+v′nruns through a complete set of representatives
of the integers modnn′.Moreover
μ^2 m=(vn′+v′n)^2 m≡(v^2 n′^2 +v′^2 n^2 )mmodnn′.
It follows that
G(mn′,n)G(mn,n′)=

∑

vmodn

∑

v′modn′

e^2 πi(mn

′ (^2) v (^2) +mn (^2) v′ (^2) )/nn′

=

∑

μmodnn′

e^2 πiμ

(^2) m/nn′
=G(m,nn′).

A deeper result is the following reciprocity formula, due to Schaar (1848):
Proposition 10For any positive integers m,n,
G(m,n)=

√

n

m

C

(^2) ∑m− 1
μ= 0
e−πiμ
(^2) n/ 2 m
,
where C=( 1 +i)/ 2.
Proof Let f:R → Cbe a function which is continuously differentiable when
restricted to the interval [0,n] and which vanishes outside this interval. Since the sum
F(t)=

∑∞

k=−∞

f(t+k)

has only finitely many nonzero terms, the functionFhas period 1 and is continuously
differentiable, except possibly for jump discontinuities whentis an integer. Therefore,