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 discuss some problems concerning the Sato-Tate type distribution and prime factorizations of norms of the Gauss-Jacobi sums. It was in our joint study by the first author with Tsuneo Arakawa on the multiple Lvalues that we first encountered with this type of character sums. The present investigation is motivated by this work ([1], but the connection to the Gauss-Jacobi sum was not mentioned there), in the hope of finding some nice arithmetic properties of the Gauss-Jacobi sums and their possible application to the theory of the multiple L-values. We have so far not been able to realize this hope but only barely begun to make a step forward to the goal. We however presume Arakawa would have been delighted to see any progress, even tiny, about the topics he once got interested in. It is thus our great pleasure to dedicate this paper to the memory of him.