It has been known since the pioneering work of Postnikov (1956) that character sums modulo prime powers $p^k$ with small $p$ and large $k$ admit bounds of much shorter length than for generic or prime moduli.

This is because they can be reduced to bounds of exponential sums with polynomials, to which Vinogradov’s method applies. The result of Postnikov was consecutively improved by Gallagher (1972) and then further generalised by Iwaniec (1974) and Chang (2014).

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 $p^k$. In turn, we improve the error term for the counting function of primes in progressions modulo $p^k$ (and in fact for a more general class of moduli).

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 $p^k$ and move it all the way up to $1$. That is, we give an asymptotic formula for the intervals of length $x$ with $p^k \le x^{1-\varepsilon}$ rather than $q \le x^{2/3-\varepsilon}$ as in the case of an arbitrary modulus $q$ (for any $\varepsilon>0$).

We will also discuss several possible extensions of these results and methods and (rather glum) perspectives for further improvements.