arXiv subjects

Jack B. Miller

Publications and source records attributed to Jack B. Miller.

5 recordsLinked to original sources

arXiv2026

A refined Malle conjecture for Heisenberg groups

Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran-Santens leading constant for many $2$-groups of nilpotency class $2$.

math.NT↗

arXiv2026

Erdős-Kac theorems for discriminants of number fields

The classical Erdős-Kac theorem gives a central limit theorem for the number of prime divisors of a random integer. We prove an analog for the number of ramified primes in a random $G$-extension of a number field when $G$ is abelian. This builds on previous work of Lemke Oliver and Thorne in the cases $G = S_d$ ($2 \le d \le 5$), and provides the first examples where local ramification events at distinct primes are not independent. We develop probability results that can be used "out of the box" to prove Erdős-Kac theorems for sequences of ideals in a number field, subject to Tauberian hypotheses involving finite sums of Euler products.

math.NT↗

arXiv2023

Patterns of primes in joint Sato--Tate distributions

For $j=1,2$, let $f_j(z) = \sum_{n=1}^{\infty} a_{j}(n) e^{2πi nz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j \ge 2$ with trivial nebentypus. For each prime $p$, let $θ_{j}(p)\in[0,π]$ be the angle such that $a_j(p) = 2p^{(k-1)/2} \cos θ_{j}(p)$. The now-proven Sato--Tate conjecture states that the angles $(θ_j(p))$ equidistribute with respect to the measure $dμ_{\mathrm ST} = \frac{2}π\sin^2θ\,dθ$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2 \subset [0,π]$, there exist infinitely many bounded gaps between the primes $p$ such that $θ_1(p) \in I_1$ and $θ_2(p) \in I_2$. We also prove a common generalization of the bounded gaps with the Green--Tao theorem.

math.NT↗

arXiv2023

Extending the support of $1$- and $2$-level densities for cusp form $L$-functions under square-root cancellation hypotheses

The Katz-Sarnak philosophy predicts that the behavior of zeros near the central point in families of $L$-functions agrees with that of eigenvalues near 1 of random matrix ensembles. Under GRH, Iwaniec, Luo and Sarnak showed agreement in the one-level densities for cuspidal newforms with the support of the Fourier transform of the test function in $(-2, 2)$. They increased the support further under a square-root cancellation conjecture, showing that a ${\rm GL}(1)$ estimate led to additional agreement between number theory and random matrix theory. We formulate a two-dimensional analog and show it leads to improvements in the two-level density. Specifically, we show that a square-root cancellation of certain classical exponential sums over primes increases the support of the test functions such that the main terms in the $1$- and $2$-level densities of cuspidal newforms averaged over bounded weight $k$ (and fixed level $1$) converge to their random matrix theory predictions. We also conjecture a broad class of such exponential sums where we expect improvement in the case of arbitrary $n$-level densities, and note that the arguments in [ILS] yield larger support than claimed.

math.NT↗

arXiv2023

Benfordness of Measurements Resulting from Box Fragmentation

We make progress on a conjecture made by [DM], which states that the $d$-dimensional frames of $m$-dimensional boxes resulting from a fragmentation process satisfy Benford's law for all $1 \leq d \leq m$. We provide a sufficient condition for Benford's law to be satisfied, namely that the maximum product of $d$ sides is itself a Benford random variable. Motivated to produce an example of such a fragmentation process, we show that processes constructed from log-uniform proportion cuts satisfy the maximum criterion for $d=1$.

math.PR↗