The reciprocals of some characteristic 2 "theta series"

TL;DR

This paper investigates the properties of certain theta-series in characteristic 2, proving zero-density results for coefficients in specific cases using modular forms theory, and discusses broader conjectures and implications.

Contribution

It introduces new theta-series in characteristic 2 and proves zero-density results for their coefficients in specific arithmetic progressions using modular forms.

Findings

Coefficients of certain theta-series have density zero in specific cases

Modular forms theory can be used to analyze these theta-series

Conjectures about the density of coefficients are discussed and partially supported

Abstract

Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases, for n restricted to certain arithmetic progressions, it is true. We prove such zero-density results using the theory of modular forms, and speculate about what may be true in general.