A problem of D. H. Lehmer in short intervals. I

A problem of D. H. Lehmer suggests to study the number of integers, each of which has different parity from its multiplicative inverse modulo $q$. We obtain an asymptotic formula for the number of such integers in very short intervals as long as $q$ is squarefree and has good factorizations, using arithmetic exponent pairs to estimate incomplete Kloosterman sums. This improves an early result of Z. Zheng in the above special cases. We also prove a prime-variable version with the aid of an estimate for incomplete Kloosterman sums in prime variables, due to Bourgain.