Jacobi Sum

# Jacobi sum

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Description:
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J([itex]chi[/itex], [itex]psi[/itex]) for Dirichlet characters [itex]chi[/itex], [itex]psi[/itex] modulo a prime number p, defined by

J([itex]chi[/itex], [itex]psi[/itex]) = Σ [itex]chi[/itex](a)[itex]psi[/itex](1 &minus; a),

where the summation runs over all residues a = 2, 3, ..., p &minus; 1 mod p (for which neither a nor 1 &minus; a is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy. Jacobi sums J can be factored generically into products of powers of Gauss sums [itex]g[/itex]. For example, when the character [itex]chipsi[/itex] is nontrivial, J([itex]chi[/itex], [itex]psi[/itex]) = [itex]g(chi)[/itex][itex]g(psi)[/itex]/[itex]g(chipsi)[/itex], analogous to the formula for the beta function in terms of gamma functions. Since the nontrivial Gauss sums [itex]g[/itex] have absolute value p<sup>1/2</sup>, it follows that J([itex]chi[/itex], [itex]psi[/itex]) also has absolute value p<sup>1/2</sup> when the characters...

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