数论

An explicit cubic Ramanujan--Sato formula for $1/\pi$ on $\Gamma_0(2)^+$ at $D=-163$ is presented. The construction produces a very small cubic CM parameter, giving about $15.01$ decimal digits of geometric contraction per term. This is…

Let \((a_n)_{n\ge1}\subset\mathbb{N}\) be a lacunary sequence, \(a_{n+1}\ge q a_n\) for \(q>1\). For \(x\in\mathbb{T}\), we study the maximal empty circular gap \(G_N(x)\) of the finite orbit \(\{a_1x,\ldots,a_Nx\}\). We prove that, for…

For a positive integer $h$, let $R_{A,h}(n)$ denote the number of ordered representations $n=s_1+\cdots+s_h$ with all $s_i\in A$. Let \[ B=\{0\}\cup\{m\ge 1:\text{ the base-4 expansion of }m\text{ begins with }1\text{ or }2\}. \] Shallit…

Here, we show that if $m\ge 5$ is fixed and odd, then there are only finitely many Carmichael numbers of the form $2^np^m+1$ for positive integers $n$ and prime $p$.

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbf{Q}_p$. For a smooth representation $\pi$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different…

We prove an asymptotic formula for a weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers. To do this, we introduce L-functions for weight $\frac {1}{2} $ sesquiharmonic Maass forms of moderate growth and prove a…

The Padovan sequence $\{P_{m}\}_{m\ge 0}$ is a ternary recurrence sequence with companion polynomial $X^{3}-X-1$ and initial conditions $P_{0}=P_{1}=P_{2}=1$. The Perrin sequence $\{R_{m}\}_{m\ge 0}$ is defined by the same companion…

We classify Boolean cubic forms in ten variables up to GL(10,2)-equivalence. The catalog contains all 3691560 nonzero orbits. For every orbit we provide a representative with small monomial count, the stabilizer order, and the alternating…

Let $P_s(n)$ denote the $n$-th $s$-gonal number. Consider the Diophantine equation $P_{s}(n) = t^{m}$ for integers $n, s, t$ and $m > 2$. All solutions to this equation are known for $m>2$ and $s\in\{3,5,6,8,10,20\}$. Here we extend these…

The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map's…

Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\). We call $ \delta(G,H)=\min_T |D(T)|…

We study the coefficients of Ramanujan's third order mock theta function \[ \rho(q)=\sum_{m\geq 0} \frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n. \] Numerical evidence suggests the striking sign…

In this paper we count the number of common values shared by two linear recurrence sequences, whose characteristic polynomials are a generalized Ankeny-Brauer-Chowla polynomial and its reciprocal. More precisely, we show that these…

For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the…

Slater's list of Rogers-Ramanujan type identities remains a central source of striking series-product formulas in the theory of partitions and basic hypergeometric series. Although many of these identities admit elegant analytic proofs…

We introduce various cohomological obstructions for smooth integral varieties over $p$-adic function fields. We show that the unramified obstruction is the finest one among obstructions arising from arithmetic dualities. We also construct…

Assuming the Generalized Riemann Hypothesis, we establish upper bounds of conjectural order of magnitude for shifted moments of the Dedekind zeta function associated with a finite Galois extension. This improves results of Milinovich and…

Given an ideal in a number field, it is desirable in many situations to find two elements that generate the ideal over the ring of the integers of the field. Existing algorithms are either randomized, or impractical at cryptographic sizes.…

We construct canonical extensions of $p$-adic shtukas on integral models of toroidal compactifications of abelian-type Shimura varieties with quasi-parahoric levels at any prime number $p$. More precisely, we define the notion of a log…

In this article we study the Iwasawa invariants of Bertolini--Darmon theta elements in the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ for weight two modular forms $f\in S_2(\Gamma_0(N))$. We cover both the…