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

Source: 1966 Swedish Mathematical Competition p5

March 21, 2021
limitlatticecombinatorial geometry

Problem Statement

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?
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