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:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“At Timon’s villalet us pass a day,
Where all cry out,What sums are thrown away!’”
—Alexander Pope (1688–1744)
“It is open to question whether the highly individualized characters we find in Shakespeare are perhaps not detrimental to the dramatic effect. The human being disappears to the same degree as the individual emerges.”
—Franz Grillparzer (1791–1872)