Mean square of zeta function, circle problem and divisor problem revisited

Abstract

This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple variant related to Section 4 in [BW17] leads to the following improvements of Theorem 3 in [BW17]

A problem with the proofs of Propositions 2 and 3 (a gap or fault in the reasoning used to claim that the expression in (3.9) is dominated by that in (3.11)); a similar problem with the proof of Proposition $1^{\prime }$ (it is hard to justify the particular application of the Bourgain-Guth reduction theory implicit in a paragraph above Proposition $1^{\prime }$). Theorems 1, 2 and 3 lose their status as theorems

Cited by in corpus (12)

• Second Moments in the Generalized Gauss Circle Problem
• The Weyl formula for planar annuli
• On an Analogue Of the Gauss Circle Problem For the Heisenberg Groups
• Sums of squares in real quadratic fields and Hilbert modular groups
• On a generalization of a theorem of Popov
• Variance of lattice point counting in some special shells in ${\mathbb{R}}^d$
• Quantitative equidistribution of eigenfunctions for toral Schrödinger operators
• A summation involving the number of divisors function and the GCD function
• Lattice points in spherical segments
• A sharp square function estimate for the cone in ${\mathbb{R}}^3$
• Cardinality of a floor function set
• A note on the Weyl formula for balls in ${\mathbb{R}}^d$