Sums of reciprocals modulo composite integers

In 1938, as part of a wider study, Emma Lehmer derived a set of four related congruences for certain sums of reciprocals over various ranges, modulo squares of odd primes. These were recently extended to congruences modulo squares of positive integers n, with certain restrictions on n. In this paper we characterize those excluded n for which the congruences still hold, and find the correct reduced moduli in the cases in which the congruences do not hold.