Editing Quadratic Gauss sum

In [[number theory]], '''quadratic Gauss sums''' are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex [[exponential function]] with coefficients given by a quadratic character; for a general character, one obtains a more general [[Gauss sum]]. These objects are named after [[Carl Friedrich Gauss]], who studied them extensively and applied them to [[quadratic reciprocity|quadratic]], [[cubic reciprocity|cubic]], and [[biquadratic reciprocity|biquadratic]] reciprocity laws.

== Definition ==

For an odd [[prime number]] {{mvar|p}} and an integer {{mvar|a}}, the '''quadratic Gauss sum''' {{math|''g''(''a''; ''p'')}} is defined as
: <math> g(a;p) = \sum_{n=0}^{p-1}\zeta_p^{an^2},</math>
where <math>\zeta_p</math> is a [[Primitive nth root of unity|primitive {{mvar|p}}th root of unity]], for example <math>\zeta_p=\exp(2\pi i/p)</math>.
Equivalently, 
: <math>g(a;p) = \sum_{n=0}^{p-1}\big(1+\left(\tfrac{n}{p}\right)\big)\,\zeta_p^{an}.</math>

For {{mvar|a}} divisible by {{mvar|p}}, and we have <math>\zeta_p^{an^2}=1</math> and thus
: <math> g(a;p) = p.</math>

For {{mvar|a}} not divisible by {{mvar|p}}, we have <math>\sum_{n=0}^{p-1} \zeta_p^{an} = 0</math>, implying that
: <math>g(a;p) = \sum_{n=0}^{p-1}\left(\tfrac{n}{p}\right)\,\zeta_p^{an} = G(a,\left(\tfrac{\cdot}{p}\right)),</math>
where
: <math>G(a,\chi)=\sum_{n=0}^{p-1}\chi(n)\,\zeta_p^{an}</math>
is the [[Gauss sum]] defined for any character {{mvar|''χ''}} modulo {{mvar|p}}.

== Properties ==

* The value of the Gauss sum is an [[algebraic integer]] in the {{mvar|p}}th [[cyclotomic field]] <math>\mathbb{Q}(\zeta_p)</math>.
* The evaluation of the Gauss sum for an integer {{mvar|a}} not divisible by a prime {{math|''p'' > 2}} can be reduced to the case {{math|''a'' {{=}} 1}}:
:: <math> g(a;p)=\left(\tfrac{a}{p}\right)g(1;p). </math>

* The exact value of the Gauss sum for {{math|''a'' {{=}} 1}} is given by the formula:<ref>M. Murty, S. Pathak, The Mathematics Student Vol. 86, Nos. 1-2, January-June (2017), xx-yy ISSN: 0025-5742 https://mast.queensu.ca/~murty/quadratic2.pdf</ref>
:: <math> g(1;p) =\sum_{n=0}^{p-1}e^\frac{2\pi in^2}{p}=
\begin{cases} 
(1+i)\sqrt{p} & \text{if}\ p\equiv 0 \pmod 4, \\ 
\sqrt{p} & \text{if}\ p\equiv 1\pmod 4, \\ 
0 & \text{if}\ p \equiv 2 \pmod 4, \\
i\sqrt{p} & \text{if}\ p\equiv 3\pmod 4. 
\end{cases}</math>

; Remark
In fact, the identity
: <math>g(1;p)^2=\left(\tfrac{-1}{p}\right)p</math>
was easy to prove and led to one of Gauss's [[proofs of quadratic reciprocity]]. However, the determination of the ''sign'' of the Gauss sum turned out to be considerably more difficult: Gauss could only establish it after several years' work. Later, [[Peter Gustav Lejeune Dirichlet|Dirichlet]], [[Leopold Kronecker|Kronecker]], [[Issai Schur|Schur]] and other mathematicians found different proofs.

== Generalized quadratic Gauss sums ==

Let {{math|''a'', ''b'', ''c''}} be [[natural numbers]]. The '''generalized quadratic Gauss sum''' {{math|''G''(''a'', ''b'', ''c'')}} is defined by

:<math>G(a,b,c)=\sum_{n=0}^{c-1} e^{2\pi i\frac{a n^2+bn}{c}}</math>.

The classical quadratic Gauss sum is the sum {{math|''g''(''a'', ''p'') {{=}} ''G''(''a'', 0, ''p'')}}.

; Properties

*The Gauss sum {{math|''G''(''a'',''b'',''c'')}} depends only on the [[residue class]] of {{math|''a''}} and {{math|''b''}} modulo {{math|''c''}}.
*Gauss sums are [[multiplicative function|multiplicative]], i.e. given natural numbers {{math|''a'', ''b'', ''c'', ''d''}} with {{math|[[greatest common divisor|gcd]](''c'', ''d'') {{=}} 1}} one has

::<math>G(a,b,cd)=G(ac,b,d)G(ad,b,c).</math>

:This is a direct consequence of the [[Chinese remainder theorem]].

*One has {{math|''G''(''a'', ''b'', ''c'') {{=}} 0}} if {{math|gcd(''a'', ''c'') > 1}} except if {{math|gcd(''a'',''c'')}} divides {{math|''b''}} in which case one has
::<math>G(a,b,c)= \gcd(a,c) \cdot G\left(\frac{a}{\gcd(a,c)},\frac{b}{\gcd(a,c)},\frac{c}{\gcd(a,c)}\right)</math>.

:Thus in the evaluation of quadratic Gauss sums one may always assume {{math|gcd(''a'', ''c'') {{=}} 1}}.

*Let {{math|''a'', ''b'', ''c''}} be integers with {{math|''ac'' ≠ 0}} and {{math|''ac'' + ''b''}} even. One has the following analogue of the [[quadratic reciprocity]] law for (even more general) Gauss sums<ref>Theorem 1.2.2 in B. C. Berndt, R. J. Evans, K. S. Williams, ''Gauss and Jacobi Sums'', John Wiley and Sons, (1998).</ref>
::<math>\sum_{n=0}^{|c|-1} e^{\pi i \frac{a n^2+bn}{c}} = \left|\frac{c}{a}\right|^\frac12 e^{\pi i \frac{|ac|-b^2}{4ac}} \sum_{n=0}^{|a|-1} e^{-\pi i \frac{c n^2+b n}{a}}</math>.

*Define
::<math> \varepsilon_m = \begin{cases} 1 & \text{if}\ m\equiv 1\pmod 4 \\ i & \text{if}\ m\equiv 3\pmod 4 \end{cases}</math>
:for every odd integer {{math|''m''}}. The values of Gauss sums with {{math|''b'' {{=}} 0}} and {{math|gcd(''a'', ''c'') {{=}} 1}} are explicitly given by

::<math>G(a,c) = G(a,0,c) =
\begin{cases}
0 & \text{if}\ c\equiv 2\pmod 4 \\ 
\varepsilon_c \sqrt{c} \left(\dfrac{a}{c}\right) & \text{if}\ c\equiv 1\pmod 2 \\ 
(1+i) \varepsilon_a^{-1} \sqrt{c} \left(\dfrac{c}{a}\right) & \text{if}\ c\equiv 0\pmod 4.
\end{cases}</math>

:Here {{math|({{sfrac|''a''|''c''}})}} is the [[Jacobi symbol]]. This is the famous formula of [[Carl Friedrich Gauss]].

* For {{math|''b'' > 0}} the Gauss sums can easily be computed by [[completing the square]] in most cases. This fails however in some cases (for example, {{math|''c''}} even and {{math|''b''}} odd), which can be computed relatively easy by other means. For example, if {{math|''c''}} is odd and {{math|gcd(''a'', ''c'') {{=}} 1}} one has

::<math>G(a,b,c) =  \varepsilon_c \sqrt{c} \cdot \left(\frac{a}{c}\right) e^{-2\pi i \frac{\psi(a) b^2}{c}},</math>

:where {{math|''ψ''(''a'')}} is some number with {{math|4''ψ''(''a'')''a'' ≡ 1 (mod ''c'')}}. As another example, if 4 divides {{mvar|c}} and {{mvar|b}} is odd and as always {{math|gcd(''a'', ''c'') {{=}} 1}} then {{math|''G''(''a'', ''b'', ''c'') {{=}} 0}}. This can, for example, be proved as follows: because of the multiplicative property of Gauss sums we only have to show that {{math|''G''(''a'', ''b'', 2<sup>''m''</sup>) {{=}} 0}} if {{math|''n'' > 1}} and {{math|''a'', ''b''}} are odd with {{math|gcd(''a'', ''c'') {{=}} 1}}. If {{mvar|b}} is odd then {{math|''an''<sup>2</sup> + ''bn''}} is even for all {{math|0 ≤ ''n'' < ''c'' − 1}}. For every {{mvar|q}}, the equation {{math|''an''<sup>2</sup> + ''bn'' + ''q'' {{=}} 0}} has at most two solutions in {{math|'''<math>\mathbb{Z}</math>'''/2<sup>''n''</sup>'''<math>\mathbb{Z}</math>'''}}. Indeed, if <math>n_1</math> and <math>n_2</math> are two solutions of same parity, then <math>(n_1 - n_2)(a(n_1 + n_2) +b) = \alpha 2^m</math> for some integer <math>\alpha</math>, but <math>(a(j_1 + j_2) +b)</math> is odd, hence <math>j_1 \equiv j_2 \pmod{2^m}</math>. {{huh|date=July 2022}}<!-- Is this Z/(2^(mZ)) or Z/((2^m)Z) which would just be 1/2^n? Use LaTeX markup to disambiguate. --> Because of a counting argument {{math|''an''<sup>2</sup> + ''bn''}} runs through all even residue classes modulo {{mvar|c}} exactly two times. The [[geometric sum]] formula then shows that {{math|''G''(''a'', ''b'', 2<sup>''m''</sup>) {{=}} 0}}.

*If {{mvar|c}} is an odd [[square-free integer]] and {{math|gcd(''a'', ''c'') {{=}} 1}}, then

::<math>G(a,0,c) = \sum_{n=0}^{c-1} \left(\frac{n}{c}\right) e^\frac{2\pi i a n}{c}.</math>

:If {{mvar|c}} is not squarefree then the right side vanishes while the left side does not. Often the right sum is also called a quadratic Gauss sum.

*Another useful formula

::<math>G\left(n,p^k\right) = p\cdot G\left(n,p^{k-2}\right)</math>

:holds for {{math|''k'' ≥ 2}} and an odd prime number {{mvar|p}}, and for {{math|''k'' ≥ 4}} and {{math|''p'' {{=}} 2}}.

==See also==

*[[Gauss sum]]
*[[Gaussian period]]
*[[Kummer sum]]
*[[Landsberg–Schaar relation]]

==References==
{{reflist}}
*{{cite book | last1 = Ireland | last2 = Rosen | title = A Classical Introduction to Modern Number Theory | publisher = Springer-Verlag | year = 1990 | isbn=0-387-97329-X }}
*{{cite book | first1 = Bruce C. | last1 = Berndt | first2 = Ronald J. | last2 = Evans | first3 = Kenneth S. | last3 = Williams | title = Gauss and Jacobi Sums | publisher = Wiley and Sons | year = 1998 | isbn=0-471-12807-4 }}
*{{cite book | first1 = Henryk | last1 = Iwaniec | first2 = Emmanuel | last2 = Kowalski | title = Analytic number theory | publisher = American Mathematical Society | year = 2004 | isbn=0-8218-3633-1}}

[[Category:Cyclotomic fields]]