Editing Quadratic residue (section)

===Dirichlet's formulas===

The first of these regularities stems from [[Peter Gustav Lejeune Dirichlet]]'s work (in the 1830s) on the [[class number formula|analytic formula]] for the [[Class number (number theory)|class number]] of binary [[quadratic form]]s.<ref>{{harvnb|Davenport|2000|pp=8&ndash;9, 43&ndash;51}}. These are classical results.</ref> Let ''q'' be a prime number, ''s'' a complex variable, and define a [[Dirichlet L-function]] as
:<math>L(s) = \sum_{n=1}^\infty\left(\frac{n}{q}\right)n^{-s}. </math>

Dirichlet showed that if ''q'' ≡ 3 (mod 4), then
:<math>L(1) = -\frac{\pi}{\sqrt q}\sum_{n=1}^{q-1} \frac{n}{q} \left(\frac{n}{q}\right) > 0.</math>

Therefore, in this case (prime ''q'' ≡ 3 (mod 4)), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, ..., ''q'' &minus; 1 is a negative number.
<blockquote>
For example, modulo 11,

:'''1''', 2, '''3''', '''4''', '''5''', 6, 7, 8, '''9''', 10 (residues in '''bold''')

:1 + 4 + 9 + 5 + 3 = 22, 2 + 6 + 7 + 8 + 10 = 33, and the difference is &minus;11. </blockquote>

In fact the difference will always be an odd multiple of ''q'' if ''q'' > 3.<ref>{{harvnb|Davenport|2000|pp=49&ndash;51}}, (conjectured by [[Carl Gustav Jacob Jacobi|Jacobi]], proved by Dirichlet)</ref> In contrast, for prime ''q'' ≡ 1 (mod 4), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, ..., ''q'' &minus; 1 is zero, implying that both sums equal <math>\frac{q(q-1)}{4}</math>.{{cn|date=November 2020}}

Dirichlet also proved that for prime ''q'' ≡ 3 (mod 4),
:<math>L(1) = \frac{\pi}{\left(2-\left(\frac{2}{q}\right)\right)\!\sqrt q}\sum_{n=1}^\frac{q-1}{2}\left(\frac{n}{q}\right) > 0.</math>
This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (''q'' &minus; 1)/2.

<blockquote>For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, and 5), but only one nonresidue (2).</blockquote>

An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.<ref>{{harvnb|Davenport|2000|p=9}}</ref>