Abstract
The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to$P_{k}(n)^{2}$, where$P_{k}(n)$is the discrepancy between the volume of the$k$-dimensional sphere of radius$\sqrt{n}$and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including$\sum P_{k}(n)^{2}e^{-n/X}$and the Laplace transform$\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions$k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums$\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral$\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.
Highlights
Let rk(m) denote the number of integer k-tuples (n1, n2, . . . , nk) such that n21 + · n 2 k = m, and let Sk (n )denote the sum of rk (m ) for m n, Sk(n) = rk (m )
The sum in Theorem 1.1 can be considered as an integral of a step function, and we study the difference between this integral and the continuous Laplace transform
We prove a second moment result without smoothing
Summary
Let rk(m) denote the number of integer k-tuples (n1, n2, . . . , nk) such that n21. rk (m ). Log(4π ) , where a0 is the constant term in the Laurent series for the meromorphic continuation of We prove this theorem in the remainder of this section. The function D(s, Pk × Pk) is otherwise analytic in the right half-plane Re s > (3 − k)/2 save for finitely many poles at nonpositive integers and, for k > 3, an additional simple pole at s = (5 − k)/2 with residue given by (5.5). In this case, noting that r4(m)/8 is multiplicative and comparing Euler products shows that r4(m) ms
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