Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1806.08480

Optimal estimates for an average of Hurwitz class numbers

Shingo Sugiyama·Masao Tsuzuki
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we give an optimal estimate of an average of Hurwitz class numbers. As an application, we give an equidistribution result of the family $\{\frac{t}{2q^{ν/2}} \ | \ ν\in \mathbb{N}, t \in \mathbb{Z}, |t|<2q^{ν/2}\}$ with $q$ prime, weighted by Hurwitz class numbers. This equidistribution produces many asymptotic relations among Hurwitz class numbers. Our proof relies on the resolvent trace formula of Hecke operators on elliptic cusp forms of weight $k\ge 2$.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shingo Sugiyama, Masao Tsuzuki. 2018-06-22. Optimal estimates for an average of Hurwitz class numbers. https://doi.org/10.1007/s11139-019-00146-z

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

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.

math.NT↗