The following article offers an explanation of the relationship of Jacobi and Gauss sums to Fermât and Artin-Schreier curves which is an analogue of the proof of Stickleberger's theorem. 1. Correspondences. The connection between cubic Fermât curves and cubic Jacobi sums was first observed by Gauss [G], who used it to study such sums. That one can compute the number of points on a Fermât curve ...

Two new expansions for partial sums of Gauss’ triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function p(n) and for the partition function pod(n) counting the partitions of n with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.

Two inequalities for weighted sums of powers are established. Applications to Jacobian elliptic functions and Legendre’s elliptic integrals of the first kind are presented. Some known and new inequalities for circular and hyperbolic functions are obtained. Applications to certain iterative means including Gauss’ arithmetic-geometric mean and Schwab-Borchardt mean are included. Mathematics subje...

We show that twisted monomial Gauss sums modulo prime powers can be evaluated explicitly once the power is sufficiently large.