In 2004, Montgomery and Soundararajan showed (conditionally) that the distribution of the number of primes
in appropriately sized intervals is approximately Gaussian and has a somewhat smaller variance than you might
expect from modeling the primes as a purely random sequence. Their work depends on evaluating sums of certain
arithmetic constants that generalize the twin prime constant, known as singular series. In particular, these sums
exhibit square-root cancellation in each term if they have an even number of terms, but if they have an odd number
of terms, there should be slightly more than square-root cancellation. I will discuss sums of singular series with
an odd number of terms, including tighter bounds for small cases and the function field analog. I will also explain
how this problem is connected to a simple problem about adding fractions.