It has been known since the pioneering work of Postnikov (1956) that character sums modulo prime powers
pk with small p and large k admit bounds of much shorter length than for
generic or prime moduli.
In a joint work with Bill Banks we modify the scheme and, using several ideas of Korobov (1974), reduce the
problem to estimating bivariate exponential sums which can be treated via a double applications of the MVT.
This can be done in a much simpler and stronger way than univariate sums, used by the previous authors.
This allows us to extend the zero-free region for L-functions modulo pk and
improve the error term for the counting function of primes in progressions modulo pk.
A similar approach has also be used, in a joint work with Kui Liu and Tianping Zhang, to prove power cancellations
among very short sums of Kloosterman sums modulo a prime power. As an application, we break Selberg's 2/3-barrier
for the average value of the divisor function in arithmetic progressions modulo pk and move it
all the way up to 1.