On the Gauss-Hecke sums

On Gauss-Jacobi sums

In this paper, we introduce a kind of character sum which simultaneously generalizes the classical Gauss and Jacobi sums, and show that this “Gauss-Jacobi sum” also specializes to the Kloosterman sum in a particular case. Using the connection to the Kloosterman sums, we obtain in some special cases the upper bound (the “Weil bound”) of the absolute values of the Gauss-Jacobi sums. We also discu...

Some Results on Gauss Sums

is again a Dirichlet character modulo n. In fact, the set of Dirichlet characters modulo n is again a multiplicative group, called the dual group of (Z/nZ)× and denoted ̂ (Z/nZ)×. The identity element of the dual group, mapping every element of (Z/nZ)× to 1, is the trivial character modulo n, denoted 1n or just 1 when n is clear, Since (Z/nZ)× is a finite group the values taken by any Dirichlet ...

Note on the Quadratic Gauss Sums

Let p be an odd prime and {χ(m) = (m/p)}, m = 0,1, . . . ,p − 1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ modp which are defined in terms of the Legendre symbol (m/p), (m,p)= 1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sumsG(k;p) are equal to the Gauss sums G(k,χ) that correspond to this ...

Elliptic Gauss Sums and Hecke L - values at s = 1 Dedicated to Professor Tomio

where p (> 3) is a rational prime such that p ≡ 3 (mod 4) and p ≡ 1 (mod 4), respectively ; ( p ) is the Legendre symbol and h(−p) is the class number of the quadratic field Q( √−p). The formulas are apparently related to the Dirichlet L-values at s = 1. To get a typical elliptic Gauss sum, we have only to replace the Legendre symbol by the cubic or the quartic residue character, and the trigon...

Half Gauss Sums