On the behavior of the function $P(x)$ on short intervals

Let $P(x)$ be an error term in the circle problem. In the talk, we introduce some known and new results concerning the connection between upper bound estimates for $\bigl|P(x\pm\sigma)-P(x)\bigr|$ and $\int_{x-\sigma}^{x+\sigma}|P(u)|^{p}\,du$ ($1\ll \sigma \ll x^{1-\varepsilon}$, $p>1$) with the pointwise estimates of the type $|P(x)|=O\bigl(x^{1-\delta}\bigr)$.