arXiv subjects

Michael Kural

Publications and source records attributed to Michael Kural.

5 recordsLinked to original sources

arXiv2025

The geometry of secondary terms in arithmetic statistics

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field $\mathbb{F}_q(t)$ of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of discriminant equal to $q^{2N}$ is $c_1 q^{2N} - c_2^{i} q^{5N/3} + O_{\varepsilon}\left(q^{(3/2+\varepsilon)N}\right)$, where $c_1$ and $c_2^{i}$ are explicit constants and $c_2^{i}$ only depends on $N\pmod{3}$. This builds on the work of Bhargava-Shankar-Tsimerman and Taniguchi-Thorne, who proved the existence of a secondary term for the count of cubic extensions of $\mathbb{Q}$ with bounded discriminant. Our approach uses a parametrization of Miranda and Casnati-Ekedahl, which can be seen as a geometric version of the classical parametrization by binary cubic forms used by Davenport-Heilbronn. This allows us to count and sieve for smooth curves embedded in Hirzebruch surfaces, in the same spirit as Zhao and Gunther.

math.NT↗

arXiv2019

Patterns of primes in the Sato-Tate conjecture

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $\mu_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $I\subseteq [-1, 1]$, let $p_{I,n}$ denote the $n$th prime such that $a_E(p)/(2\sqrt{p})\in I$. We show $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty$ for all $m\ge 1$ for "most" intervals, and in particular, for all $I$ with $\mu_{ST}(I)\ge 0.36$. Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.

math.NT↗

arXiv2019

M\"obius formulas for densities of sets of prime ideals

We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $\delta(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{\mu(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = \delta(S), \] where $\mu(\mathfrak{a})$ is the generalized M\"obius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of minimal norm lying in $S$. Our result can be applied to give formulas for densities of various sets of prime numbers, including those lying in a Sato-Tate interval of a fixed elliptic curve, and those in Beatty sequences such as $\lfloor\pi n\rfloor$.

math.NT↗

arXiv2018

Dedekind Sums with Even Denominators

Let $S(a,b)$ denote the normalized Dedekind sum. We study the range of possible values for $S(a,b)=\frac{k}{q}$ with $\gcd(k,q)=1$. Girstmair proved local restrictions on $k$ depending on $q\pmod{12}$ and whether $q$ is a square and conjectured that these are the only restrictions possible. We verify the conjecture in the cases $q$ even, $q$ a square divisible by $3$ or $5$, and $2\le q\le 200$ (the latter by computer), and provide progress towards a general approach.

math.NT↗

arXiv2018

Fuss-Schröder Paths and Rooted Plane Forests

We describe a bijection between $(k,k)$-Fuss-Schröder paths of type $λ$ and certain rooted plane forests with $n(k+1)+2$ vertices. This yields a recursion which allows us to analytically enumerate the number of large $(k,r)$-Fuss-Schröder paths of type $λ$, solving an open question posed by An, Jung, and Kim. Furthermore, we generalize the concept of $(k,r)$-Fuss-Schröder paths to $(k,S)$-Fuss-Schröder paths, in which $r$ can take any value in a given set $S$, and enumerate these paths as well.

math.CO↗