数论

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…

Let $D$ be the ring of $S$-integers in a global field and $\da$ its profinite completion. We propose a profinite version of the Bateman--Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations.…

Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong S\'{a}rk\"{o}zy-type result: if $A\subseteq [N]$ has no nonzero…

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means.…

We resolve Erdos Problem 731 under the explicit dyadic-regularity formalization of "reasonable." Let $A(n)$ be the least positive integer not dividing $\binom{2n}{n}$. On dyadic intervals $X\le n<2X$, put $L=\log(2X)$ and ${\mathcal…

Let $X/K$ be a smooth projective variety defined over a number field and $f:X\to X$ be a morphism defined over $K$. Assuming there exists a point in $X(K)$ whose $f$-orbit is Zariski dense in $X$ and up to replacing $K$ by a finite…

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function $\sum_{x\le n \le x+y} f(n)$ over short intervals $[x, x+y]$, where $y \rightarrow \infty$ but $y=o(x)$. We show that with appropriate…

Let $A$ be an abelian variety of dimension $g$ over a finite field $\mathbf{F}_q$. We show that if $q$ is sufficiently large relative to $g$, the $g$ point counts $\#A(\mathbf{F}_{q^i})$ for $1 \leq i \leq g$ determine the zeta function of…

Let $k\ge 2$ be fixed. We study the distribution modulo one of the $n^k$ sums \begin{equation*} \sqrt{a_1} + \cdots + \sqrt{a_k}, \qquad 1\le a_1, \dots, a_k \le n, \end{equation*} counted with multiplicity. For \begin{equation*} S(h,n) =…