On the Dirichlet divisor problem in short intervals

Abstract

We present several new results involving $\Delta (x+U)-\Delta (x)$, where $U=o(x)$ and $$\Delta (x):=\sum _{n\le x}d(n)-x\log x-(2\gamma -1)x$$ is the error term in the classical Dirichlet divisor problem.

20 pages