Covering systems with the sum of the reciprocals of the moduli close to $1$

TL;DR

This paper proves that in finite covering systems, if the minimum modulus exceeds 4, then the sum of the reciprocals of the moduli is necessarily less than 1, confirming a longstanding conjecture.

Contribution

The paper establishes that a minimum modulus greater than 4 guarantees the sum of reciprocals in covering systems is bounded away from 1, resolving a problem posed by Davenport.

Findings

Minimum modulus greater than 4 implies sum of reciprocals is less than 1

Confirms a conjecture by Erdos and Selfridge from 1973

Provides a proof for a problem in covering systems theory

Abstract

In 1952, H. Davenport posed the problem of determining a condition on the minimum modulus $m_{0}$ in a finite distinct covering system that would imply that the sum of the reciprocals of the moduli in the covering system is bounded away from $1$. In 1973, P. Erdos and J. Selfridge indicated that they believed that $m_{0} > 4$ would suffice. We provide a proof that this is the case.