数论

Let $Q = \{q_n\}_{n \ge 1}$ be a sequence of integers with $q_n \ge 2$ for all $n \in\mathbb{N}$. For any real number $x \in [0,1)$, it can be expanded into the following infinite series: $$x =\frac{\varepsilon_1(x)}{q_1}+…

We consider the problem of estimating numerically integrals of the shape $$ \int_P \frac{dt}{t_1 \dotsb t_k} $$ where $P \in {\mathbb R}_{>0}^k$ is a convex polytope, $t=(t_1,\dotsc, t_k)$ and $d t$ is the Lebesgue measure. This type of…

Cohen, Manin, and Zagier recovered the Rankin-Cohen bracket for modular forms from an action of the modular group on pseudodifferential operators whose coefficients are holomorphic functions on the Poincar\'e upper half plane. We…

In this paper, we study the congruence $\binom{qn}{n} \equiv q^n \pmod n$ for a prime base $q$. Motivated by the OEIS sequence \seqnum{A080469} and the conjectural existence of infinitely many ternary solutions of the form $n=3^t p$, we…

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density. Erd\H{o}s later constructed an infinite arithmetic progression of odd integers none of which…

Let $k\geq 2$ be a positive integer. It is known that the set of visible lattice points from the origin in $\mathbb{Z}^k$ has a translation bounded pure point diffraction spectrum. We investigate these properties for sets of points…

Let $k\ge4$ and $j\ge2$ be integers with $j$ even, and let $f$ be a primitive elliptic cusp form of weight $2k+j-2$ for $\mathrm{SL}_2(\mathbb{Z})$. We study congruences between a Hermitian Klingen--Eisenstein lift associated with $f$ and…

Let $p\ge 3$ 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 $l$ of the period is the (multiplicative) order of $b$ mod $p$. In the…

Let $G$ be a split reductive group over a $p$-adic field. We give germ expansions of Kloosterman integrals for $G$. As an application, we prove that Bessel distributions are regular for all generic representations on $G$ provided that…

This article proves non-trivial estimates for a bilinear sum involving the Fourier coefficients of a Hecke-holomorphic or Hecke-Maass cusp form for $\mathrm{SL}(2,\mathbb{Z})$. As corollaries, we draw interesting results related to…

Let $F$ be a number field. Given finitely many $F$-valued points on a commutative algebraic group defined over $F$, a question of interest to number theorists is the determination of the group of their linear relations. In this article, we…

We prove that, for almost all even integers $N>0$, the set of Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ has a level of distribution $1/6$. As applications, we show that almost all even integers $N>0$ can be written as the sum of two…

We record a Weyl-positive reduction and certificate framework for the Riemann phase kernel associated with the even Riemann kernel $\Phi$. The manuscript does not present a complete proof of the Riemann hypothesis. Its immediate analytic…

For a fixed integer $k \ge 3$, we study the multiplicative functions $f\colon\mathbb{N}\to\mathbb{C}$ satisfying \[ f\Bigl(\sum_{i=1}^{2k} x_i^2\Bigr) = \sum_{j=1}^{k} f\bigl(x_{2j-1}^2 + x_{2j}^2\bigr) \] for all positive integers…

Let $G/\mathbb{Q}_p$ be a connected, split, reductive group over $\mathbb{Q}_p$. In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the…

Let $V \subset \mathbb{A}^2(\mathbb{C})$ be an algebraic curve such that $\mathrm{deg} X \neq \mathrm{deg} Y$, where $X, Y$ denote the coordinate functions on $\mathbb{A}^2(\mathbb{C})$ restricted to $V$. We prove there exists an…

Let $K=\mathbb{Q}(\sqrt{D})$ with $D>1$ squarefree, and let $\varepsilon_+$ be the totally positive fundamental unit of $\mathcal{O}_K$. B. M. Kim proved in 2000 that the octonary diagonal form \[…

Let $K/\mathbb Q_p$ be finite and let $f\in\mathcal O_K[X]$ be monic, of degree at least two, with $f'(X)\in\mathfrak m_K\mathcal O_K[X]$, equivalently $\bar f\in k[X^p]$. For a compatible inverse branch $f(t_{n+1})=t_n$ with…

A brief overview of results concerning the connection between the Hilbert-Polya conjecture and the Riemann hypothesis about the Riemann zeta function, some new results on p-adic quantum computing, quantum entanglement based on lattice spin…

Let $p$ be an odd prime number and let $\overline{\mathbb{F}}_p$ be a fixed algebraic closure of the finite field of order $p$. Let $K$ be a global function field of characteristic different from $p$ and let $G_{K}$ be the absolute Galois…