Chowla–Mordell theorem

Chowla–Mordell theorem In mathematics, the Chowla–Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and published independently by Sarvadaman Chowla and Louis Mordell, autour de 1951.

En détail, si {style d'affichage p} is a prime number, {style d'affichage chi } a nontrivial Dirichlet character modulo {style d'affichage p} , et {style d'affichage G(chi )=sum chi (un)zeta ^{un}} où {displaystyle zeta } is a primitive {style d'affichage p} -th root of unity in the complex numbers, alors {style d'affichage {frac {g(chi )}{|g(chi )|}}} is a root of unity if and only if {style d'affichage chi } is the quadratic residue symbol modulo {style d'affichage p} . The 'if' part was known to Gauss: the contribution of Chowla and Mordell was the 'only if' direction. The ratio in the theorem occurs in the functional equation of L-functions.

References Gauss and Jacobi Sums by Bruce C. Berndt, Ronald J.. Evans and Kenneth S. Williams, Wiley-Interscience, p. 53. Catégories: Cyclotomic fieldsZeta and L-functionsTheorems in number theory