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.