JacobiSymbol

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed m

Values at fixed points

General characteristics

Domain and analyticity

((n/m)) is a nonanalytical function of n and m which is defined for n∈ℤ and nonnegative odd m.

Symmetries and periodicities

Quasi-permutation symmetry

Periodicity

No periodicity

Series representations

Other series representations

Transformations

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Identities

Functional identities

Representations through equivalent functions

With related functions

Inequalities

Zeros

Theorems

The Gauss reciprocity law

((p/q))((q/p))==(((-1))^(((p-1)/2)((q-1)/2)))/;p,q>2∧p,q∈ℙ

Eigenvector of the discrete Fourier transform

For the odd prime p (p mod 4≠3) the list {((0/p)),((1/p)),…,(((p-1)/p))} is an eigenvector of the discrete Fourier transform with eigenvalue one:

((n/p))==(1/(Sqrt[p]))(∑_(k=0)^(p-1))((k/p))(e^(2π i k n/p)).

Gauss' sums

Gauss sums can be calculated using the Jacobi symbol

(∑_(Underscript[k=1, gcd(k,p)==1])^p)(e^((2 h π i (k^2))/p))==((h/p)) (i^(((((p-1)/2)))^2)) (Sqrt[p])-1/;          p∈ℙ,p>2∧h∈(SuperPlus[ℕ]).

History

– L. Euler (1729, published in 1785)
– A.-M. Legendre (1785, 1798) introduced the symbol ((n/m)) for solving quadratic congruences
– C.F. Gauss (1796)
– C. G. J. Jacobi (1837)

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