Properties of Gauss Sums of Dirichlet Characters
The Gauss sum of a Dirichlet character modulo N is
If χ is moreover primitive, then
in particular, it is non-zero. More generally, if N0 is the conductor of χ and χ0 is the primitive Dirichlet character modulo N0 that induces χ, then the Gauss sum of χ is related to that of χ0 by
where μ is the Möbius function. Consequently, G(χ) is non-zero precisely when N/N0 is squarefree and relatively prime to N0. Other relations between G(χ) and Gauss sums of other characters include
where χ is the complex conjugate Dirichlet character, and if χ′ is a Dirichlet character modulo N′ such that N and N′ are relatively prime, then
The relation among G(χχ′), G(χ), and G(χ′) when χ and χ′ are of the same modulus (and χχ′ is primitive) is measured by the Jacobi sum J(χ, χ′). Specifically,
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