Risk Theory and Geometric Sums

A Geometric Perspective on the Riemann Zeta Function’s Partial Sums

The Riemann Zeta Function, ζ(s), is an important complex function whose behavior has implications for the distribution of the prime numbers among the natural numbers. Most notably, the still unsolved Riemann Hypothesis, which states that all non-trivial zeros of the zeta function have real part one-half, would imply the most regular distribution of primes possible in the context of current theo...

Positive trigonometric sums and applications

Some new positive trigonometric sums that sharpen Vietoris’s classical inequalities are presented. These sharp inequalities have remarkable applications in geometric function theory. In particular, we obtain information for the partial sums of certain analytic functions that correspond to starlike functions in the unit disk. We also survey some earlier results with additional remarks and comments.

Dedekind sums: a combinatorial-geometric viewpoint

Dedekind sums : a combinatorial - geometric viewpoint Matthias Beck and Sinai Robins

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational poly-topes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer...

Artin-Schreier Curves, Exponential Sums, and Coding Theory

Lachaud, G., Artin-Schreier curves, exponential sums, and coding theory, Theoretical Computer Science 94 (1992) 295-310. This is a survey of some results recently obtained on the distribution of the weights of some classical linear codes on the one hand, such as the dual of the Melas code, and the geometric BCH codes discovered by Goppa (subfield subcodes of Goppa codes) on the other hand. Thes...