Quadratic Gauss Sum - Definition

Definition

Let p be an odd prime number and a an integer. Then the Gauss sum mod p, g(a;p), is the following sum of the pth roots of unity:

g(a;p) =\sum_{n=0}^{p-1}e^{2{\pi}ian^2/p}=\sum_{n=0}^{p-1}\zeta_p^{an^2}, \quad \zeta_p=e^{2{\pi}i/p}.

If a is not divisible by p, an alternative expression for the Gauss sum (with the same value) is

Here is the Legendre symbol, which is a quadratic character mod p. An analogous formula with a general character χ in place of the Legendre symbol defines the Gauss sum G(χ).

Read more about this topic:  Quadratic Gauss Sum

Famous quotes containing the word definition:

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)