On the Gauss circle problem over smooth numbers

Abstract: The Gauss circle problem concerns with the evaluation of $\sum_{n \leq x}r(n)$, where $r(n)$ denotes the number of representations of $n$ as sums of two squares and $x \geq 2$. Let $Ψ_G(x,y)$ denote the sum of $y$-smooth numbers below $x$ weighted by $r(n)$. In this paper, we evaluate $Ψ_G(x,y)$ asymptotically for certain ranges of $x \geq y \geq 2$.

• The paper introduces a novel asymptotic evaluation of ΨG(x, y) by extending the Gauss circle problem framework to y-smooth integers using saddle point methods.
• It adapts classical techniques like Rankin's trick and Hildebrand-Tenenbaum analysis to derive explicit error bounds, achieving an error term of O(1/u).
• The results refine previous asymptotic estimates and have significant implications for analytic number theory and cryptographic applications through improved sieve bounds.

Asymptotic Evaluation of the Gauss Circle Problem over Smooth Numbers

Introduction

The paper "On the Gauss circle problem over smooth numbers" (2604.23918) delivers a rigorous asymptotic analysis of a weighted counting function $\Psi_G(x, y)$, where $x \geq y \geq 2$, summing the number of integer representations as sums of two squares over $y$-smooth integers up to $x$. The classical Gauss circle problem estimates $\sum_{n \leq x} r(n)$ with $r(n)$ being the number of such representations, but this work generalizes it to the domain of smooth numbers, which are integers with all prime factors $\leq y$. By combining analytic number theory techniques and exploiting saddle point methods inspired by Hildebrand and Tenenbaum [HT86], the paper establishes new ranges and sharper uniformity for the asymptotic formulae of $\Psi_G(x, y)$.

Background, Problems, and Prior Work

The Gauss circle problem has been analyzed extensively with error terms $O(x^\theta)$, with $\theta=131/208+\varepsilon$ being the best known, and conjectured to be as low as $x \geq y \geq 2$0. For smooth numbers, Goswami [Goswami24] provided

$x \geq y \geq 2$1

for $x \geq y \geq 2$2, $x \geq y \geq 2$3, uniformly in $x \geq y \geq 2$4 and $x \geq y \geq 2$5. Here, $x \geq y \geq 2$6 is the Dickman function. Prior analytic frameworks, such as [HT86], analyze counting functions over smooth numbers using Rankin's trick and saddle point analysis based on partial derivatives of the logarithm of truncated zeta functions.

Main Results

The paper extends this methodology to the weighted sum $x \geq y \geq 2$7 defined by

$x \geq y \geq 2$8

where $x \geq y \geq 2$9 is the maximal prime factor of $y$0. The core results can be summarized as follows:

Saddle Point Asymptotics

For $y$1, $y$2, and large $y$3:

$y$4

where $y$5 is the unique saddle point solution to

$y$6

with $y$7 the relevant Euler product and $y$8 its $y$9-th partial derivative. The formula holds uniformly for prescribed ranges of $x$0 and $x$1, broadening the previously known regularity.

Explicit Asymptotics of Key Parameters

For $x$2 and $x$3 large,

$x$4

and the main term can be expressed (using $x$5 as defined in the paper) as:

$x$6

with $x$7 the Euler constant. This matches the Dickman function asymptotics at large $x$8, confirming consistency with classical results but valid for strictly larger parameter regimes.

Methodology and Technical Highlights

The analytic approach starts from Rankin's trick and is formalized via saddle point analysis—identifying the minimum of $x$9 for $\sum_{n \leq x} r(n)$0. The paper proves existence and uniqueness of $\sum_{n \leq x} r(n)$1 via convexity arguments and evaluates the relevant Euler products. The treatments also cover error bounds on sieve weights, fine partial summations over primes, and careful handling of Dirichlet characters (notably $\sum_{n \leq x} r(n)$2) supporting the two-square representation function $\sum_{n \leq x} r(n)$3.

Perron's formula enables passage to contour integration and localization near the saddle point. Various nontrivial lemmas establish uniform asymptotics for derivatives of $\sum_{n \leq x} r(n)$4, control of local factors, and sub-asymptotic error term bounds (notably exponential in $\sum_{n \leq x} r(n)$5). The consistency with prior work ([HT86]) is achieved by appropriately adapting the local analysis from the unweighted case to the weighted case.

Numerical Results and Claims

The most significant claim is the sharp uniform asymptotic for $\sum_{n \leq x} r(n)$6 over smooth numbers, especially for much larger ranges of $\sum_{n \leq x} r(n)$7 than previously handled. The formulae exhibit accurate control of the main term and error, validated by extensive theoretical lemmas and logical consistency with known results. Bounds are explicit in terms of $\sum_{n \leq x} r(n)$8, $\sum_{n \leq x} r(n)$9, and parameter derivatives, and error rates are given as $r(n)$0 alongside exponential suppression by $r(n)$1.

Implications and Theoretical Consequences

This work provides effective asymptotics for a weighted counting function central to analytic number theory and the theory of friable (smooth) numbers. The explicit formulae enable quantitative analysis of arithmetic functions over smooth integers, supporting better sieve bounds and analytic estimates in the study of quadratic forms. The theoretical consistency with Dickman function asymptotics and confirmation of convexity criteria in the saddle point analysis reinforce the robustness of the analytic framework.

Practically, the results offer precise estimates and bounds for computations involving representations as sums of two squares among smooth numbers. In cryptographic applications—where such smoothness conditions feature prominently—the results may inform performance estimates for factoring algorithms and related analytic tools.

Speculation on Future Research

Further investigation could focus on extending the weighted asymptotic analysis to sums of higher powers, other quadratic forms, or more general representation functions. Analyses of short intervals, uniformity in $r(n)$2, or the distribution in arithmetic progressions could deepen the theoretical understanding. Additionally, adapting these results to probabilistic models of smooth numbers or connections with L-functions and automorphic forms may reveal new insights. Numerical experiments may also be constructed to empirically test the uniformity and sharpness of the bounds derived.

Conclusion

The paper establishes a rigorous framework for evaluating the Gauss circle problem over smooth numbers, generalizing prior results and widening the domain of asymptotic validity. The technical machinery is robust and the results are explicit, theoretically consistent, and broadly applicable in analytic number theory. These findings reinforce the power of saddle point and analytic techniques in evaluating weighted sums over friable integers and lay groundwork for further advancements in the distribution of arithmetic functions over restricted sets.