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,
Read more about this topic: Gauss Sum
Famous quotes containing the words properties of, properties, sums and/or characters:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)
“Socialist writers are made of sterner stuff than those who only let their characters steeplechase through trouble in order to come out first in the happy ending of moral uplift.”
—Christina Stead (1902–1983)