数论

We offer some comments on series involving the M$\ddot{o}$bius function which approximate sums over primes. To accomplish this, we utilize the derivative of the Gram series by applying Riemann-Stieltjes integration. We offer a new formula…

In this paper, we study the special values of Rankin-Selberg L-functions as a continuation of [LLS24]. Utilizing the modular symbol approach, we prove the rationality and period relations for some critical values of Rankin-Selberg…

Every natural number greater than $2$ can be written as the sum of a prime and a square-free number, and recent work has imposed additional divisibility conditions on the square-free number. We overcome limitations in these works to prove…

In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries…

Let $p$ be a prime and $\mathcal{K}$ be an imaginary quadratic field. In this paper we generalize a recent construction of a new type of $p$-adic $L$-function and $p$-adic Waldspurger formula by Andreatta-Iovita for $p$ non-split in…

We show that the category of smooth representations of GL_2(Q_p) on p-power torsion modules localizes over a certain projective scheme, and give some applications.

We compare the Iwasawa invariants of fine Selmer groups of $p$-adic Galois representations over admissible $p$-adic Lie extensions of a number field $K$ to the Iwasawa invariants of ideal class groups along these Lie extensions. More…

We show that the action of Hecke operators away from $p$ on the space of ($p$-adic) overconvergent modular forms is ($p$-adically) locally analytic in a certain sense. As a corollary, the action of the Hecke algebra can be extended…

Let $a$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by…

Applying the $q$-Zeilberger algorithm, we establish a unified $q$-analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a…

We prove that for each fixed $m \ge 2$, there are only finitely many disjoint covering systems with minimum modulus at least $3$ in which precisely one modulus is repeated, namely the largest modulus, and it occurs exactly $m$ times.

Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$,…

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most…

We have been studying Eratosthenes sieve as a discrete dynamic system, obtaining exact models for the relative populations for small gaps (currently gaps $g \le 82$) in the cycle of gaps ${\mathcal G}(p^\#)$ at each stage of the sieve. The…

We explore families of pairs of quadratic polynomials $f(x)=x^2+c\in \mathbb{Q}$ and $a\in \mathbb{Q}$ with $a$ being a strictly preperiodic point of $f$ to provide infinitely many new examples for which the associated arboreal Galois…

Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindel\"of hypothesis. We obtain asymptotic…

In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to…

We study integral points on the quadratic twists $E_D : y^2 = x^3+D^2Ax+D^3B$ of a fixed elliptic curve $E : y^2 = x^3+Ax+B$ over $\overline{Q}$. For sufficiently large squarefree positive integers $D$, we prove that the number of integral…

Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where…