On the Sphere Problem

On Sums of Three Squares

“…Let r 3 (n) be the number of representations of n as a sum of three squares (counting signs and order). It was conjectured by Hardy and proved by Bateman [1] that (1) r 3 (n) = 4πn 1/2 S 3 (n),…”

Lattice points in the 3-dimensional torus

“…The crucial step in the proof of Proposition 4.1 is a variation of the arguments of [CC12] that are a generalization of [CI95] and give a bound for…”

“…The exponent comes from a diagonal term and then it seems unlikely to be improved in the context of the classical approaches based on exponential sums. If one could count with precision (beyond the limits of the harmonic analysis) points close to the boundary of the scaled theory then one could parallel the arguments of [CI95] (improved in [HB99]) or [CCU09] to go beyond 4/3. The multiplicative harmonics (Dirichlet characters) and the automorphic harmonics employed in these papers apparently cannot be adapted to the torus.…”

A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds

“…In fact, for M = R n /Z n , we see that (1) {u → e(m · u) : m ∈ Z n } is a basis for the eigenfunctions of the Laplace operator ∆ = − Nr. P20847-N18.…”

“…There exists a vast literature on this particular subject: We only refer to the works of Huxley [7], [8], Hafner [4], and Soundararajan [17] for the planar case, for the papers by Chamizo & Iwaniec [1], HeathBrown [5], and Tsang [18] for dimension n = 3 , and to the monographs of Walfisz [20], and Krätzel [14], [15], as well as to the recent, quite comprehensive, survey article [9].…”