Questions tagged [gauss-circle-problem]

Consider a generalization of the Gauss circle problem and let $$ N(r):= \#\lbrace (x,y,z)\in \Bbb Z^3_{\gt 0}\vert \ln^2(x)+\ln^2(y)+\ln^2(z)\le\ln(r) \rbrace $$ I found that $$N(r)=\left(\frac{2\pi}{...

Quoting myself from an earlier post: Pólya's orchard problem asks for which radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of ...

You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...

For the Gauss circle problem $$ R(x):=\sum_{0 \leq n \leq x} r_2(n)=\pi x+P(x), P(x)=O(x^{\frac{1}{4}+\epsilon}) $$ Gauss may not know the integral formula of the error term. $$ \int_0^X|P(x)| d x=O\...