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

A problem of D. H. Lehmer suggests to study the number of integers, each of which has different parity from its multiplicative inverse modulo $q$. For large prime $q$, we obtain an asymptotic formula for the number of such integers up to $N$, where $N$ is a bit smaller than $q^{1\mathrm{/}2}$. This beats the barrier $q^{1\mathrm{/}2}$ in the prime modulus case. An estimate for the second moment of the error term on average over $q$ is also established. The main inputs are estimates for several bilinear forms with Kloosterman fractions.