数论

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_q$ of $q$ elements. A polynomial in $\mathbb{F}_q[t]$ is called $m$-smooth (or $m$-friable) if all its irreducible factors are of degree at most $m$. In this…

Let $\mathscr{O}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathop{\mathrm{Spec}} \mathscr{O}_K$ endowed with the canonical log structure. We show that the generic fiber…

With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree d polynomials f that are ordered by height. This allows us to establish averaged versions of the Bateman-Horn conjecture, the…

This work is a probabilistic study of the 'primes' of the Cram\'er model. We prove that there exists a set of integers $\mathcal S$ of density 1 such that \begin{equation}\liminf_{ \mathcal S\ni n\to\infty} (\log n)\mathbb{P} \{S_n\…

We establish omega results for the divisor and circle problems that are conjecturally sharp, while also determining the sign of the large values obtained. This improves on the work of Soundararajan and on the subsequent independent…

Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo…

We investigate a fifty-year-old conjecture of Erd\H{o}s and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is ``close'' to $n$: that is, bigger…

We utilize the structure of quasiautomorphic forms over a Hecke triangle group to define a mapping from a quasiautomorphic form to a vector-valued automorphic form (vvaf). This kind of vvaf we call a Hecke vector-form. First we supply a…

We consider the asymptotics of sums of the form $$ \frac1{F_n^\sigma} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(\pi m/F_n)}\right|^\sigma} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(\pi F_{n-1}m/F_n)}\right|^\sigma} $$ where $(F_n)_{n \in…

Let $K$ be a totally real number field of odd degree. Let $l \geq 5$ be a prime with $l \nmid [K:\mathbb{Q}]$ and $\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1$. We prove that if $2$ is inert in $K$, $l$ is non-Wieferich, i.e., $2^{l-1} \not\equiv…

We continue our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy}. We establish asymptotic formulas for each term of the trace formula when summing over $n<X$, using arbitrary smooth test…

In a work of Heath-Brown, it is proved that in the Pilz divisor problem, the normalized error term $\Delta_3(x)$ has a distribution function. In this paper, we prove an analogue of this result in the setting of GL(3). For a given self-dual…

This paper studies the problem of discrepancy estimates for pseudorandom vectors constructed by the elliptic curve congruential generator, particularly in the non-translational case. Two families of results are obtained. First, in a…

We study the distribution of eigenvalues of Haar-random matrices over $\mathbb{Z}_p$ among algebraic extensions of $\mathbb{Q}_p$. Our results give $p$-adic analogues of the real-eigenvalue counting results of Edelman-Kostlan-Shub for the…

The Connes-van Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter $c$ that controls the primes $p\leq c$ entering the operator, has a ground state whose Fourier-Mellin zeros provably lie on the critical line; whether…

In this article, we study admissible representations of even unitary groups over local fields, where the quadratic extension is ramified, with invariant vectors under the action of the stabilizer of a unimodular lattice and some properties…

Let $p>3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $q$ of the period is the (multiplicative) order of $b$ mod $p$. In the case…

We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois…

We study the one-level density of zeros for a family of $\Gamma_1(q)$ $L$-functions. Assuming GRH, we are able to extend the support of the Fourier transform of the test function to $\left(-\frac{8}{3},\frac{8}{3}\right)$ and verify the…

Let $G$ be a reductive group over a local field $F$ and let $\rho:{}^LG \to \mathrm{GL}_{V_{\rho}}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ng\^o conjectured that one has a…