MathDB

5

Part of 1966 Swedish Mathematical Competition

Problems(1)

limit with f(r), no of lattice points inside the circle radius r

Source: 1966 Swedish Mathematical Competition p5

3/21/2021
Let f(r)f(r) be the number of lattice points inside the circle radius rr, center the origin. Show that lim⁡r→∞f(r)r2\lim_{r\to \infty} \frac{f(r)}{r^2} exists and find it. If the limit is kk, put g(r)=f(r)−kr2g(r) = f(r) - kr^2. Is it true that lim⁡r→∞g(r)rh=0\lim_{r\to \infty} \frac{g(r)}{r^h} = 0 for any h<2h < 2?
limitlatticecombinatorial geometry