Averaging on thin sets of diagonal forms

TL;DR

This paper studies the behavior of diagonal forms over thin sets, analyzing their average asymptotic formulas and error terms, and develops methods to compute p-adic densities of zeros for large primes.

Contribution

It introduces new analysis of average asymptotic formulas for diagonal forms on thin sets and provides an effective approach to compute p-adic densities of zeros.

Findings

Analysis of asymptotic formula evolution

Effective computation of p-adic densities

Second moment analysis for zero densities

Abstract

We investigate one-dimensional families of diagonal forms, considering the evolution of the asymptotic formula and error term. We then discuss properties of the average asymptotic formula obtained. The subsequent second moment analysis precipitates an effective means of computing $p$-adic densities of zeros for large primes $p$.