Kloosterman's sum

Kloosterman's sum may be defined, for integers u, v, n, by where denotes a primitive n-th root of unity. The following function uses residue class rings and cyclotomic fields to compute such sums.

> Kloosterman := func< u, v, n |
>     &+[z^(IntegerRing() ! (u*x+v*x^-1)) : x in F | IsUnit(x) ]
>         where z is RootOfUnity(n, CyclotomicField(n))
>         where F is ResidueClassRing(n) >;

We can verify the multiplicative identity for some random example:

> s := Kloosterman(3, 11, 12);
> t := Kloosterman(3, 7, 25);
> s, t;
2
-5*zeta_25^2 + 5*zeta_25^3 - 5*zeta_25^7 - 5*zeta_25^12 - 5*zeta_25^17
> u := Minimize(Kloosterman(3, 7*12^2+11*25^2, 12*25));
> u;
-10*zeta_25^2 + 10*zeta_25^3 - 10*zeta_25^7 - 10*zeta_25^12 - 10*zeta_25^17
> u eq s*t;
true;

(The Minimize function used above returns a given element of a cyclotomic field in the smallest possible cyclotomic field.)

If one wishes to view the Kloosterman sum as a complex number, one could use ! to do the conversion (where, by convention, ). In the above example:

> ComplexField(12) ! u;
14.57937257 - 0.00000013 i

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