Easy answer to Gauss's Circle Problem?

Question

From reading the Wikipedia article it seems that the unsolved bit of Gauss's Circle Problem deals with the error term $E(r)$ in the equation $N(r) = \pi r^2 + E(r)$, where $r$ is the radius of the circle and $N(r)$ is the amount of integer lattice points within the circle. Farther down the page, an exact formula is given for $N(r):$ $$N(r) = 1 + 4\sum_{i=0}^{\infty} (\lfloor \frac{r^2}{4i+1}\rfloor - \lfloor \frac{r^2}{4i+3}\rfloor)$$ Why is it wrong (I'm assuming it's wrong) to say that $$E(r) = (1 + 4\sum_{i=0}^{\infty} (\lfloor \frac{r^2}{4i+1}\rfloor - \lfloor \frac{r^2}{4i+3}\rfloor)) - \pi r^2$$? Wouldn't this be exact, thus a lower and upper bound simultaneously for $E(r)$, with which the problem deals? Have I misunderstood $E(r)$?

Answer 1

Your expression is correct, but it does not solve the problem. The problem is to provide a bound on how $E(r)$ depends on $r$. We cannot read off a limit on how large $E(r)$ can be. In the article it says that $E(r)$ grows at least as fast as $r^{1/2}$ and no faster than $r^{131/208}$