Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.04632

A decades-long breakthrough in zero-density estimates and primes in short intervals

Caroline L. Turnage-Butterbaugh
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Riemann Hypothesis (RH) asserts that every nontrivial zero of the Riemann zeta-function has real part equal to $1/2$. A zero-density theorem provides evidence towards RH by bounding the number of zeros of the zeta-function with real part greater than $1/2$. In 2024, Larry Guth and James Maynard announced a new zero-density theorem which, for a key location in the critical strip, strengthens previous work of Ingham and is the first such improvement in over 80 years. This expository paper places this remarkable achievement in the context of the rich history of zero-density theorems and explores its implications on the distribution of primes in short intervals.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Caroline L. Turnage-Butterbaugh. 2026-07-06. A decades-long breakthrough in zero-density estimates and primes in short intervals. https://arxiv.org/abs/2607.04632

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

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

math.NT↗