Odd moments and adding fractions

Abstract We prove near‐optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application, we prove near‐optimal upper bounds for the average of the refined singular series in the Hardy–Littlewood conjectures concerning the number of prime ‐tuples for odd. The main new ingredient is a near‐optimal upper bound for the number of solutions to when is odd, with and restrictions on the size of the numerators and denominators, which is of independent interest.