Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.0299

The distribution of smooth numbers in arithmetic progressions

K. Soundararajan
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a wide range of $x$ and $y$ we show that ${\Cal S}(x,y)$, the set of integers below $x$ composed only of prime factors below $y$, is equidistributed in the reduced residue classes $\pmod q$ for all $q<y^{4\sqrt{e}-ε}$. This improves earlier work of Granville; any improvement of this range of $q$ would have interesting consequences for Vinogradov's conjecture on the least quadratic non-residue. For larger ranges of $q$ we prove the existence of a large subgroup of the group of reduced residues such that ${\Cal S}(x,y)$ is equidistributed within cosets of that subgroup.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

K. Soundararajan. 2007-07-02. The distribution of smooth numbers in arithmetic progressions. https://arxiv.org/abs/0707.0299

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$, revisited

Let $r(N)$ be the number of representations of $N$ as a sum of two squares. For every integer $n\ge 2$ we prove an asymptotic formula for $\sum_{x\equiv e \bmod 2}Φ(x/M)\,r(n^2+x^2)$, that is, for the number of integral solutions of $x_1^2+x_2^2-x_3^2=n^2$ with $x_3\equiv e \bmod 2$ in a range $|x_3|\ll M$, with smooth weight $Φ$. The formula has an explicit secondary term and a power saving in the error term, and it is non-trivial as soon as $M\ge n^{5/6+\varepsilon}$, that is, $M\ge D^{1/2-1/12+\varepsilon}$ with $D=n^2$. The leading term is $\frac{4}π(\log n)\intΦ(x/M)\,dx$, independently of the arithmetic of $n$; this corrects the main term stated in the published version (Ramanujan J. 63 (2024), 293-337). The proof factorises $n+ix_3$ in $\mathbb{Z}[i]$, which leads to Kloosterman sums. These are estimated by Weil's bound, combined with a Plancherel argument that exploits the curvature of the underlying quadric. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.

math.NT↗
arXiv2026

Bessel Periods on $U(2,1) \times U(1,1)$, Relative Trace Formula and Non-Vanishing of Central $L$-values

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(π, π')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $π$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, π\times π')$, which are of degree twelve over ${\mathbb Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also leads, by known results, to deducing certain associated Selmer groups have rank zero.

math.NT↗
arXiv2026

Rational points on $X_0(N)^*$ when $N$ is non-squarefree

Let $N$ be a positive integer and $X_0(N)^*$ be the quotient of the modular curve $X_0(N)$ by its full group of Atkin-Lehner involutions. Elkies conjectures that the rational points on $X_0(N)^*$ are only cusps or CM points when $N$ is large enough. We establish an integrality result for the $j$-invariants of non-cuspidal rational points on $X_0(N)^*$ when $N$ is non-squarefree, representing a significant step toward resolving a key subcase of Elkies' conjecture. To this end, we prove the existence of rank-zero quotients of certain modular Jacobians $J_0(pq)$. Furthermore, in the non-squarefree case we provide a complete classification of the rational points on $X_0(N)^*$ of genus $1 \leq g \leq 5$, when they are finite. In the process we identify exceptional rational points on $X_0(75)^*$ and $X_0(147)^*$ which were not known before.

math.NT↗