[Paper Review] Smooth Sums over Smooth $k$-Free Numbers and Statistical Mechanics

Francesco Cellarosi|arXiv (Cornell University)|Apr 24, 2013

Analytic Number Theory Research23 references3 citations

TL;DR

This paper establishes an asymptotic formula for smooth sums over $k$-free integers with bounded prime factors, incorporating a complex parameter $\alpha$ and a smooth cut-off function $f$. The main result expresses the sum as a product of a power of $\log N$ and an integral involving the $\alpha$-convolution of the Dickman-de Bruijn distribution, with explicit error bounds depending on $\alpha$, $f$'s regularity, and a truncation parameter $R$. The work generalizes prior results for $k=2$, $\alpha=1$ to complex $\alpha$ and broader function classes.

ABSTRACT

We provide an asymptotic estimate for certain sums over k-free integers with small prime factors. These sums depend upon a complex parameter αand involve a smooth cut-off f. They are a variation of several classical number-theoretical sums. One term in the asymptotics is an integral operator whose kernel is the α-convolution of the Dickman-de Bruijn distribution, and the other term is explicitly estimated. The trade-off between the value of αand the regularity of f is discussed. This work generalizes the results of tow previous papers by the author and Ya.G. Sinai, where k=2 and α=1.

Motivation & Objective

• To extend previous asymptotic results on sums over $k$-free numbers to complex parameters $\alpha$ and smooth cut-offs $f$.
• To analyze the interplay between the regularity of $f$ (measured by $\eta$) and the truncation parameter $R(N)$ in controlling error terms.
• To derive an explicit asymptotic expansion involving the $\alpha$-convolution of the Dickman-de Bruijn distribution as a key component.
• To generalize earlier results for $k=2$, $\alpha=1$ to arbitrary $k \geq 2$ and $|\alpha| < 2$, under suitable regularity and decay conditions on $f$.
• To quantify the trade-off between the smoothness of $f$ and the growth rate of $R(N)$ in the error term.

Proposed method

• The sum $S_{\Omega,f}(k,\alpha;N)$ is analyzed via Fourier analysis, using the Fourier transform $\hat{f}$ of the smooth cut-off function $f$.
• A complex measure is introduced to re-express the sum in terms of characteristic functions, leading to the appearance of $\varphi^{(\alpha)}(\lambda)$, the $\alpha$-convolution of the Dickman-de Bruijn distribution.
• Error terms are estimated using integral bounds involving the exponential integral $\mathrm{Ei}$ and incomplete gamma functions $\Gamma(0,z)$, with careful asymptotic analysis as $N \to \infty$.
• The truncation parameter $R(N)$ is introduced to localize the Fourier integral, and the error $\varepsilon_N$ is decomposed into two parts: $O(\log\log N / \log N)$ and $O((\log N)^{|α| - \Re\alpha}/R^{\eta-1})$, which are controlled via $R(N)$'s growth rate.
• The method relies on splitting the sum into dyadic intervals and applying stationary phase-type estimates to oscillatory integrals arising from the exponential terms $e^{i\lambda v}$.
• The use of the function space $\mathcal{S}_\eta(\mathbb{R})$ ensures sufficient decay of $\hat{f}$ to control the error, with $\eta > |\alpha| - \Re\alpha + 1$ being a critical condition.

Experimental results

Research questions

• RQ1 How does the asymptotic behavior of smooth sums over $k$-free numbers depend on the complex parameter $\alpha$ and the smoothness of the cut-off function $f$?
• RQ2 What is the precise role of the $\alpha$-convolution of the Dickman-de Bruijn distribution in the leading term of the asymptotic expansion?
• RQ3 How do the regularity of $f$ (measured by $\eta$) and the truncation scale $R(N)$ trade off in determining the error term?
• RQ4 Can the results for $k=2$, $\alpha=1$ be generalized to arbitrary $k \geq 2$ and complex $\alpha$ with controlled error?
• RQ5 What is the asymptotic behavior of the integral term $\int_{|\lambda| \leq R} \varphi^{(\alpha)}(\lambda) \hat{f}(\lambda) \, d\lambda$ as $N \to \infty$?

Key findings

• The sum $S_{\Omega,f}(k,\alpha;N)$ admits the asymptotic expansion $S_{\Omega,f}(k,\alpha;N) = C \cdot (\log N)^\alpha \cdot \left( \int_{|\lambda| \leq R} \varphi^{(\alpha)}(\lambda) \hat{f}(\lambda) \, d\lambda + \varepsilon_N \right)$, where $C = C(k,\alpha)$ is a non-zero complex constant.
• The error term $\varepsilon_N$ satisfies $\varepsilon_N = O(\log\log N / \log N) + O((\log N)^{|α| - \Re\alpha}/R^{\eta-1})$, with both terms tending to zero as $N \to \infty$ under the given conditions.
• For $R(N) = \log N / \log\log N$, the error is $O(\log\log N / \log N)$ if $\eta > |\alpha| - \Re\alpha + 2$, and otherwise decays polynomially in $\log N$.
• For $R(N) = (\log N)^{1-\tau}$ with $0 < \tau < 1$, the error term exhibits a phase transition between two regimes depending on $\tau$ and $\eta$, with explicit decay rates in each case.
• The integral term $\int_{|\lambda| \leq R} \varphi^{(\alpha)}(\lambda) \hat{f}(\lambda) \, d\lambda$ is $O(1)$, but may decay to zero; a concrete example is given where it remains bounded away from zero.
• The results extend to smooth sums over $k$-free numbers excluding finitely many primes, allowing $|\alpha| < 3$ for odd $k$-free integers, demonstrating robustness of the method.