Visible lattice points in the sphere

Abstract

The number of visible (primitive) lattice points in the sphere of radius R is well approximated by 4 π 3 ζ ( 3 ) R 3 . We consider an integral expression involving the error term E ∗ ( R ) , which leads to E ∗ ( R ) = Ω ( R ( log R ) 1 / 2 ) . This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E ∗ ( R ) . This approach to prove Ω-results seems to be new and could be useful in other problems.