In mathematics, Brewer sums are finite

character sum In mathematics, a character sum is a sum \sum \chi(n) of values of a Dirichlet character χ '' modulo'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of quadratic ...

introduced by related to

Jacobsthal sum In mathematics, Jacobsthal sums are finite sums of Legendre symbols related to Gauss sums. They were introduced by . Definition The Jacobsthal sum is given by :\phi_n(a)=\sum_\left(\dfrac\right) where ''p'' is prime and () is the Legendre symbol ...

Definition

The Brewer sum is given by :

\Lambda_n(a) = \sum_\binom

where ''D''_''n'' is the

Dickson polynomial In mathematics, the Dickson polynomials, denoted , form a polynomial sequence introduced by . They were rediscovered by in his study of Brewer sums and have at times, although rarely, been referred to as Brewer polynomials. Over the complex num ...

(or "Brewer polynomial") given by :

D_(x,a)=2,\quad D_1(x,a)=x,  \quad D_(x,a)=xD_n(x,a)-aD_(x,a)

and () is the

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue ...

. The Brewer sum is zero when ''n'' is

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...

to ''q''²−1.

References

* * * * Number theory {{numtheory-stub